Intrepid won't start from key, only starts if starter is bumpede.

1999 Dodge Intrepid 2.7 won't start from key. Will start if you bump the starter, but won't disengage. Starter just keeps running. When I first started having problems for 2 days it would click on first couple tries and then start. On day 3 it would click or start. Figured it was the starter and bought a reman. Now its not making the clicking noise. But only starts from bumping the starter. Check the battery and relays all seem fine. What could the problem be.