Less than 1%About 6% of an answer.
(maybeI've worked out the letters in 3 squares so far. Maybe someone will find this useful while I work out the remaining 99%94%)
The bottom left corner is marked 4, so one of the words that use that corner must completely overlap another that also uses that corner. Only the beginning of words contain other words so that corner must contain T.
Words that could overlap are:
TEA - TEAM, TEAR
TEE - TEEM
Overlapping words must extend diagonally from the corner. So square 1,1 is T and square 2,2 is E. The centre square is either M or R. The top right corner must have a diagonal word that starts or ends there (it shares 3 words) so that word must either share the M/R centre square or be 3 letters. There are a number of possibilities for that word but they all require square 6,6 to be E.
The letter frequencies reveal some useful information:
T 18
E 15
A 10
R 5
H 4
I 2
C 2
L 2
U 2
M 2
K 1
D 1
G 1
Y 1
K, D, G and Y appear only once so words must be placed so that these letters don't fall on a square numbered more than 1. Similarly I, C, U, L and M appear twice each, so care must be taken not to place them on squares numbered more than 2.