1. You should sue for the value of the ring. Essentially, when the NHS Care home took the wedding ring off, they became bailees or safekeepers of the ring for your mother. So your mother is entitled to sue in law for the value of the ring back. Accordingly, you should speak with a lawyer and get proceedings issued on your mother's behalf.
2. The value for which your mother can sue is the value today. Not the value 10 years ago.
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3. In order for the Care Home to be liable it is enough that they lost the ring. You do not have to show how it was lost. So your mother can successfully sue for the value today of the ring. However, because there is no evidence of who took it, the police will not treat it as theft or get involved in bringing criminal proceedings.