What is the ruling on the tawaf and sa'i of a woman who entered into ihram while she was menstruating out of ignorance, then performed tawaf after becoming pure?

If you have performed Tawaf and Sa'i after purification and then shortened your hair, you are disengaged from your ihram and nothing else is required of you.

If you confined yourself to Tawaf and omitted Sa'i, then if you are in a state of ihram according to the view that Sa'i is a pillar [of Hajj/Umrah], you must intend to go to Mecca, perform the Sa'i, and then shorten your hair to be disengaged. If you are unable to do so, you disengage as one who is prevented (muhsar) by slaughtering a sacrificial animal (hady) and then shortening your hair. If you are unable to provide a hady, then fast ten days and then shorten your hair.

However, according to the view that Sa'i is an obligation (wajib) and not a pillar (rukn), if you have performed Tawaf and shortened your hair, you are disengaged from your ihram. However, for omitting Sa'i, a compensatory sacrifice (dam) is required, to be slaughtered in Mecca. If you are unable to provide the dam, then fast ten days.

If you performed another Umrah after the one whose Sa'i you omitted, it takes its place, and nothing else is required of you.

Regarding the ihram prohibitions you committed during that period, it is more cautious to offer a fidyah for those that fall under the category of damage, such as cutting hair and trimming nails. Some scholars, however, hold the view that nothing is required of one who committed them out of ignorance.