What is the ruling on repeatedly breaking a promise to Allah not to abandon prayer, and what must be done as expiation for that?

A servant should abide by what he has promised his Lord, especially if it is among the obligations. Breaking this promise does not entail anything [specific in terms of worldly consequence, but rather spiritual culpability]. Forsaking prayer is one of the greatest sins. Therefore, you must repent, regret what has passed, and not return to this sin.