Front load washer doesn't drain well, found broken part

I have a Kenmore HE3T washer that I bought in 2003. Up until about a year ago, we really had no problems with it. For the past year, it would periodically fill with water when not in use. I thought it was the water inlet valve. I was going to replace it, and then the washer stopped doing it. So I never got around to it.

Now it's doing the same thing. And it doesn't seem to be draining as well during cycles. Yesterday a towel was sopping wet. Today my husband opened it up to completely clean out the lint trap area. There wasn't much debris in there, but he found a little red drum and two screws laying loose in the bottom. It also looks like this little red drum was snapped off from somewhere, as the plastic is broken.

I've tried googling, and I can't seem to find any answers. What is this little plastic piece? How can I fix this?

 

I need your model number.....it's long.....like 110.xxxxxxxxxxxxxx

It looks like it may be part of the pump assembly.

http://www.searspartsdirect.com/part...shdPart=280187

 

Red part is not part of washer. Have no idea what it is. Screws could of come out of pants pocket. Can.t see how big red part is but by sze of screws not to big so probably a toy. Fill valve is causing to fill.