I think to solve the problem it would be easier to buy an used transmission.
The 5ha gear of the L251 has a "left" and a "right" part
picture
At first you have to remove the cap on the left side. Then you have to remove the 5. Gear and some other parts. Afterwards you have to release the middle part. And then you can release both gear shafts and the differential. Much work.
But for the first step, before you start to work at the gear: Try to remove the left drive shaft from the gear and check the snap ring.
(in this picture on the left side). Take some light and take a look into the differential. It seems to me that the left drive shaft is not engaged, so maybe the snap ring was broken.
A damage of the differential is very unusual.
Sorry fo my bad english...if you do not understand everythink just ask!