All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
The parties of a contract are obliged to abide by the terms and conditions stipulated in the contract. Allah, The Exalted, says (what means): {O you who have believed, fulfill (all) contracts.} [Quran 5:1]
The Prophet, sallallahu ‘alayhi wa sallam, said, "Muslims are bound by their conditions except for a condition that makes the lawful unlawful or the unlawful lawful." [At-Tirmithi: Hasan-Saheeh (good or sound)]
Hence, the reference regarding what you asked about is the contract between your father and the club and the terms that are stipulated in it. If the contract does not stipulate the continuity of the marriage bond in order to retain the membership, then there is no harm on your mother in benefiting from the club membership after her divorce.
Therefore, you need to read the contract and its terms and ask the administration of the club in order to know the situation. You are not required to inform them of your mother's divorce, as you can ask a general question without mentioning the names of the divorced couple.
The health condition of your mother, which you mentioned, does not make it permissible for her to benefit from the club if she has no right to that according to the contract and its terms.
It should be noted that the divorce renders your mother a non-Mahram (alien) woman to your father if it is irrevocable and the same thing applies if it was revocable but the ‘Iddah (waiting period) has expired. It is impermissible for him to stay alone with her or mix with her in a prohibited manner. For more benefit, please refer to fatwa 96286.
Allah knows best.