What is the depreciation value of an 11-year-old built-in kitchen?

A fitted kitchen is usually depreciated over 7 years.

Wrong. A fitted kitchen is depreciated over 10 years.

Either way: it is worth nothing anymore. At least not in the sense of a "price achievable on the open market." In the sense of a "price on which two people agree," it can still have value if the former landlady apparently wants it. Just wait for an offer, ask for a bit more, and that's it.

A tenant cannot be obliged to purchase a fitted kitchen upon moving in - but then the landlord can simply look for another tenant. Your mother now owns the kitchen, which of course probably no longer has any residual value. Just leave it in? Then the owner can have it removed and disposed of at your expense. I wouldn’t risk that. Of course, you can try to offer the kitchen for sale to the owner. Perhaps start with 200 euros; if he does not agree, ask if you can leave it in. But please make sure to get this confirmed in writing. If he insists that the kitchen must be removed, you can try using classified ads. If the appliances are still good, then sell them individually and try to give away the rest of the kitchen - or dispose of it.

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The new owner wants to have the kitchen, but the tenant does not want to give it to him.

Hello, take photos and put the kitchen on eBay Kleinanzeigen. 300€ pickup only. And that's it.

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The new owner wants to have the kitchen, but the tenant refuses to give it up to him.

True!
But I’m afraid the owner knows exactly what his rights are. And it’s simply a very bad basis if your mother wants to demand an exorbitant price that the kitchen definitely isn’t worth anymore. And the owner will know that he DOES NOT HAVE to buy the kitchen: So he can be completely relaxed – and if necessary, “get back” at your mother.