It shouldn't need a brute-force approach.

DALEY is a five-digit factor of 13, and because none of the digits are duplicated, it must have a minimum value of 10234.

This means that TOM must be greater than 10234/13 = 787.23076

Because TOM not only has no repeated digits but also no digits in common with DALEY it must lie between 789 and 987 and doesn't include the digit '1'

This in turn means that the highest value for DALEY must be 987*13 = 12831, and cannot be in the range 11000-11999 i.e. it must start with either 10 or 12. Therefore TOM must lie between either 789-845, or 926-987.

I then just knocked up a spreadsheet to display the remaining possible combinations and quickly scanned through the combinations, eliminating further blocks of numbers with duplicated digits, to find that the only solution is:

796*13 = 10348