Weld on the side gears? What are you talking about? You're not going to HAVE a differential anymore, you'll have a permanently locked rear end.
Why don't you just go to a junkyard and find another differential?
Also, if you've torn up those so much, better take your axle completely apart and clean it out and replace all the bearings or you'll be taking it apart soon enough anyway...and probably not when you're expecting it.