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I have now confirmed that...

636236 → six hundred thirty-six thousand two hundred thirty-six
636622 → six hundred thirty-six thousand six hundred twenty-two
636626 → six hundred thirty-six thousand six hundred twenty-six
636632 → six hundred thirty-six thousand six hundred thirty-two
636636 → six hundred thirty-six thousand six hundred thirty-six

... all of which have ...

$3+7+6+3+8+3+7+6+3 = 46$ length,
vowel count of $13$, and
digit count of 6,
for LS value of $46/(13+6)=2.42105263157895$

... have the highest Linguistic Stiffness scores.

Each component maximizes the ratio of length to vowels:
"Two" and "six", "twenty" and "thirty" are the highest ratio values usable in their relevant positions. And the longer numbers tend to allow maximum total length of the text for the number, but "million" with 3 vowels ends up offsetting its benefit by enough that not including numbers in the millions ends up winning.
Other numbers which substitute a "two" for a "six" or a "twenty" for a "thirty", or vice versa, would have equivalent Linguistic Stiffness.

In the interest of completeness,

There are 64 numbers with the winning LS value. They are:

222222 222226 222232 222236 222622 222626 222632 222636
226222 226226 226232 226236 226622 226626 226632 226636
232222 232226 232232 232236 232622 232626 232632 232636
236222 236226 236232 236236 236622 236626 236632 236636
622222 622226 622232 622236 622622 622626 622632 622636
626222 626226 626232 626236 626622 626626 626632 626636
632222 632226 632232 632236 632622 632626 632632 632636
636222 636226 636232 636236 636622 636626 636632 636636

I have now confirmed that...

636236 → six hundred thirty-six thousand two hundred thirty-six
636622 → six hundred thirty-six thousand six hundred twenty-two
636626 → six hundred thirty-six thousand six hundred twenty-six
636632 → six hundred thirty-six thousand six hundred thirty-two
636636 → six hundred thirty-six thousand six hundred thirty-six

... all of which have ...

$3+7+6+3+8+3+7+6+3 = 46$ length,
vowel count of $13$, and
digit count of $6$,
for LS value of $46/(13+6)=2.42105263157895$

... have the highest Linguistic Stiffness scores possible.

Each component maximizes the ratio of length to vowels:
"Two" and "six", "twenty" and "thirty" are the highest ratio values usable in their relevant positions. And the longer numbers tend to allow maximum total length of the text for the number, but "million" with 3 vowels ends up offsetting its benefit by enough that not including numbers in the millions ends up winning.
Other numbers which substitute a "two" for a "six" or a "twenty" for a "thirty", or vice versa, would have equivalent Linguistic Stiffness.

In the interest of completeness,

There are 64 numbers with the winning LS value. They are:

222222 222226 222232 222236 222622 222626 222632 222636
226222 226226 226232 226236 226622 226626 226632 226636
232222 232226 232232 232236 232622 232626 232632 232636
236222 236226 236232 236236 236622 236626 236632 236636
622222 622226 622232 622236 622622 622626 622632 622636
626222 626226 626232 626236 626622 626626 626632 626636
632222 632226 632232 632236 632622 632626 632632 632636
636222 636226 636232 636236 636622 636626 636632 636636

I have now confirmed that...

636236 → six hundred thirty-six thousand two hundred thirty-six
636622 → six hundred thirty-six thousand six hundred twenty-two
636626 → six hundred thirty-six thousand six hundred twenty-six
636632 → six hundred thirty-six thousand six hundred thirty-two
636636 → six hundred thirty-six thousand six hundred thirty-six

... all of which have ...

$3+7+6+3+8+3+7+6+3 = 46$ length,
vowel count of $13$, and
digit count of 6,
for LS value of $46/(13+6)=2.42105263157895$

... have the highest Linguistic Stiffness scores.

Each component maximizes the ratio of length to vowels:
"Two" and "six", "twenty" and "thirty" are the highest ratio values usable in their relevant positions. And the longer numbers tend to allow maximum total length of the text for the number, but "million" with 3 vowels ends up offsetting its benefit by enough that not including numbers in the millions ends up winning.
Other numbers which substitute a "two" for a "six" or a "twenty" for a "thirty", or vice versa, would have equivalent Linguistic Stiffness.

I have now confirmed that...

636236 → six hundred thirty-six thousand two hundred thirty-six
636622 → six hundred thirty-six thousand six hundred twenty-two
636626 → six hundred thirty-six thousand six hundred twenty-six
636632 → six hundred thirty-six thousand six hundred thirty-two
636636 → six hundred thirty-six thousand six hundred thirty-six

... all of which have ...

$3+7+6+3+8+3+7+6+3 = 46$ length,
vowel count of $13$, and
digit count of 6,
for LS value of $46/(13+6)=2.42105263157895$

... have the highest Linguistic Stiffness scores.

Each component maximizes the ratio of length to vowels:
"Two" and "six", "twenty" and "thirty" are the highest ratio values usable in their relevant positions. And the longer numbers tend to allow maximum total length of the text for the number, but "million" with 3 vowels ends up offsetting its benefit by enough that not including numbers in the millions ends up winning.
Other numbers which substitute a "two" for a "six" or a "twenty" for a "thirty", or vice versa, would have equivalent Linguistic Stiffness.

In the interest of completeness,

There are 64 numbers with the winning LS value. They are:

222222 222226 222232 222236 222622 222626 222632 222636
226222 226226 226232 226236 226622 226626 226632 226636
232222 232226 232232 232236 232622 232626 232632 232636
236222 236226 236232 236236 236622 236626 236632 236636
622222 622226 622232 622236 622622 622626 622632 622636
626222 626226 626232 626236 626622 626626 626632 626636
632222 632226 632232 632236 632622 632626 632632 632636
636222 636226 636232 636236 636622 636626 636632 636636

Updated

I believe that...

636236 → six hundred thirty-six thousand two hundred thirty-six
636622 → six hundred thirty-six thousand six hundred twenty-two
636626 → six hundred thirty-six thousand six hundred twenty-six
636632 → six hundred thirty-six thousand six hundred thirty-two
636636 → six hundred thirty-six thousand six hundred thirty-six

... all of which have ...

$3+7+6+3+8+3+7+6+3 = 46$ length,
vowel count of $13$, and
digit count of 6,
for LS value of $46/(13+6)=2.42105263157895$

... are going to be the highest scores.

Each component maximizes the ratio of length to vowels:
"Two" and "six", "twenty" and "thirty" are the highest ratio values usable in their relevant positions. And the longer numbers tend to allow maximum total length of the text for the number, but "million" with 3 vowels ends up offsetting its benefit by enough that not including numbers in the millions ends up winning.
Other numbers which substitute a "two" for a "six" or a "twenty" for a "thirty", or vice versa, would have equivalent Linguistic Stiffness.

I have now confirmed that...

636236 → six hundred thirty-six thousand two hundred thirty-six
636622 → six hundred thirty-six thousand six hundred twenty-two
636626 → six hundred thirty-six thousand six hundred twenty-six
636632 → six hundred thirty-six thousand six hundred thirty-two
636636 → six hundred thirty-six thousand six hundred thirty-six

... all of which have ...

$3+7+6+3+8+3+7+6+3 = 46$ length,
vowel count of $13$, and
digit count of 6,
for LS value of $46/(13+6)=2.42105263157895$

... have the highest Linguistic Stiffness scores.

Each component maximizes the ratio of length to vowels:
"Two" and "six", "twenty" and "thirty" are the highest ratio values usable in their relevant positions. And the longer numbers tend to allow maximum total length of the text for the number, but "million" with 3 vowels ends up offsetting its benefit by enough that not including numbers in the millions ends up winning.
Other numbers which substitute a "two" for a "six" or a "twenty" for a "thirty", or vice versa, would have equivalent Linguistic Stiffness.