So, when you turn the key to "START" you get a click down by the solenoid, correct? If so, don't change any more parts, just read this article:
www.mylawnmowerforum.com
Also, watch this you-tube video:
Should fix you right up. Let me know if you need any more info or have any questions.

Starter Assist Relay
***Update! Sister Site MTF Member Stripmine drew up some illustrations to help with the CORRECT way to install a Starter Assist Relay. It's with his permission that I have inserted those drawings into this submission. It really shows "how to" and especially "how NOT to". My thanks to him...
Also, watch this you-tube video:
Should fix you right up. Let me know if you need any more info or have any questions.