Is that a Briggs 125K02-0189-E1?

If I can do it, anyone can.

Bill - no, it's a 126T02-0206-B1. So far I have been unable to track down a replacement for it .

Re-powering your 20055 is reasonably easy. I didn't have any trouble re-powering mine, and I'm not a mechanic.

If you go to sites like Brand New Engines, Tulsa Engine Warehouse, and Small Engine Warehouse (among others) they have guides that can help you find a replacement Briggs engine that will work with that model.

If you want a Honda GCV160 engine instead, you'll need to order a new blade bolt, blade driver assembly, and brake cable from Toro, because, according to Toro's Master Parts Viewer, the ones on the 20055 won't fit this engine. (However, also according to the Toro Master Parts Viewer, the mounting screws for the 20055 will fit this engine.)

I ended up contacting B&S, and they directed me to a 3rdParty vendor that came back with the recommended replacement engine. It is model 126M02-1031-F1, in case anyone is interested. FedEx just dropped it off, so I'll see if it really is a plug-n-play replacement, as promised :-) My project for the weekend ... if it ever stops raining!

It was probably best going to "the horses mouth".
It appears that has a somewhat odd crank shaft.