Many of the nails used to hold down the tack strip didn't come up, and the
heads break off when I try to pull 'em out

Dremel grinder & grind them down flush ?

Don't send a boy to do a man's job. Buy a low cost angle grinder (or rent one,
it;s bound to be a bargain). It will do the whole job in under 10 minutes.
Harbor Freight sells them for less thgan $10, other outlets specializing in
Chinese tools are similar. If you then find the tool useful, add a Milwaukee or
other good brand to your collection and loan the cheapie to your neighbor. HTH

Joe