What is the ruling on one who vowed to perform supererogatory acts of obedience after his recovery, but then did not fulfill that vow?

The stated wording is not an oath or a vow; rather, it is closer to a promise. A person should fulfill their promise to observe supererogatory acts of worship (nawafil) because fulfilling promises is among the commendable virtues, especially if it pertains to obedience to Allah. If one does not fulfill it, there is no expiation required of them, but they should repent and seek forgiveness.