Is making a vow to perform missed prayers considered valid, and is it permissible to abandon it due to suffering from a severe mental illness accompanied by intense chest pain?

You are obligated to make up the obligatory prayers you missed since the day you reached puberty, in whatever way is easy for you, as long as it does not lead to negligence or dereliction. You do not have a concession to abandon making up the prayers, and you must strive to do so to the best of your ability and capacity. As for your vow, it is not considered valid by the majority of scholars because it is a foregone conclusion. However, some scholars hold that it is binding and necessitates an expiation for breaking an oath if it is not fulfilled.