"I cut the meat with a knife."
This sentence confused me for a moment, since "cut" is the same in both present be past tense in English.
ナイフでにくをきります should be accepted. There is nothing in the English sentence suggesting it is referring to the past.
I don't think that works by word ordering - you need つかって to be associated with "knife" if you're going to use it at all, but your ordering suggests you're using the meat.
I do wonder about ナイフでつかってにくをきりました, though; does that not work because 《で》is the wrong particle and it would need 《を》instead?