The extra verse seems to suggest that the answer might be
King Henry VIII
and this certainly seems to be correct, but some details remain hazy.
So, as cleverly suggested by Marius in the comments, the numbers in stanza one, at least, are counting years rather than, e.g., monarchs. So, let's see.
To start at number seventeen
One does rise so prominently
Henry VIII became king at age 17.
With thirty eight now in between
And fifty five completely
38 years after that he died at age 55.
That all seems straightforward enough. The second stanza has given me much grief and continues to do so. All those sixes of course suggest
his wives
but I can't make everything line up.
Six, six to find the one
The six wives he married to get one legitimate male heir? (The questioner confirms this, despite my doubtfulness on the grounds that he only needed three wives for that.)
Six, six but found in three
As gtwebb suggests in comments, this may simply be because "the one" was born to H8's third wife. One of the sixes here (both?) may refer not to wives but to the fact that he became Edward the Sixth.
Six, six and follows six
Might possibly be counting daughters, but to get enough we would need to include stillborn ones contrary to what seems to have been done with the sons. (But they did "follow six" in the sense of ascending the throne later than his son Edward VI.)
OP Edit: Edward VI succeeds Henry VIII
Five, but only count one
Most likely refers to his sons; presumably the "one" is the single surviving legitimate son, Edward. Now, why five? He had a bunch of stillborn sons, but too many for the count to be five. If we only count the ones who lived beyond infancy, then we have Edward (legitimate) plus Henry FitzRoy (acknowledged illegitimate) plus, er, three of { Henry Carey, John Perrot, Thomas Stukeley, Richard Edwardes }?
Or
perhaps the five are his acknowledged liveborn children. One legitimate son, Edward. (The one we are to count.) Two legitimate girls, Mary and Elizabeth. One acknowledged illegitimate son, Henry FitzRoy. And one other Henry who died in infancy.
OP Edit: References number of illegitimate sons Henry VIII allegedly had, only acknowledging one of them.