Step 1
Decide what exactly needs to be fixed. With Wassily chairs, the most common problem is the aging of the leather arms, back and seat. Over time, the leather warps, sags or comes undone. In the case of serious buckling or ripped leather, you'll have to replace the entire strap. Unless you're an experienced leather sewer who owns an industrial sewing machine, you'll need to take the chair to a leather workshop. The leather of a Wassily chair must be thick and tight to support a body's weight.
If the problems are largely cosmetic, such as stains or scratches in the leather from pets or belts, you can dye the leather with a color enhancement kit. A cheap solution is to rub shoe polish into the leather until it covers any stained areas, but this is risky.
Step 2
Try to determine the edition of chair you own. Knoll stamps its versions with Breuer's signature plus an identification and authenticity number. If you don't see a stamp of authenticity, that doesn't mean the chair isn't authentic. Particularly if it's older, it may have been manufactured by Gavina (or by Thonet, though unlikely).
Step 3
Use strong leather glue to glue together the two pieces of leather, if the stitching has worn through enough that the leather has separated into its two separate parts. Clamp the leather pieces together after applying glue and let it dry overnight.
Step 4
Restitch any undone seams using waxed thread and an upholstery needle. Forcing the needle through the leather can be difficult, so you may want a thimble or a hard and heavy object to help push the needle through. You can also ask a cobbler to help you.