Need help with Craftsman push mower

I have thouroughly cleaned the carb (twice), changed oil, air filter, gas, spark plug, etc. Literally everything. However, When I removed the case, I found this part that was just kind of free floating. See attached picture. Seems like there might be a missing spring or some sort of connection there. I even found the exact diagram for the model on the Sears website but it did not show this piece. Can't find anything about it online. Any help is greatly appreciated! Thanks

EDIT: It will run for three seconds with a spray of starting fluid but that's it.

That piece is part of the Briggs AutoChoke system. The part will be located in the diagram which has the muffler in it. There are some U-tube videos on how to adjust. There should be a small spring on the carb end of the plastic air vane arm. It should keep the choke closed until the engine starts and then the air vane will open the choke. The metal part has a thermistor which will keep the choke open once the engine warms up.

This'll help you out!
https://youtu.be/YAqVcdZOZE4?t=113

Thank you, Lawn King and ScrubCadet! After checking all of the things that could relate to the autochoke, I realized a spring was in the wrong place. However, that was not the main problem. After taking the carb apart again, I found that the main jet was CLOGGED. Bad. I assumed I got it when I sprayed it out twice before. I was wrong. Lol. Thank you both very much for your help. I am starting to work on machines a lot, so there is a good chance I will be back! Haha. Take care!!

Glad it you got it fixed up! yep, Gas-o-hol (ethanol gas) wreaks havoc on carbs if you let it sit for a bit, i.e. clogging up the main jet.
:thumbsup::thumbsup: