Quick answer
Do I need to make up missed prayers after my period?
No. Missed prayers during ḥayḍ and nifās are not made up — this is the consensus of the four madhhabs. Only missed fasts are made up.
No. Prayers missed during ḥayḍ or nifās are not made up. This is the unanimous position of the four Sunni madhhabs.
ʿĀʾishah (raḍiyallāhu ʿanhā) was asked why a menstruating woman makes up the fast but not the prayer, and she responded: 'That used to happen to us and we were ordered to make up the fast but were not ordered to make up the prayer.' (Sahih Muslim)
This is a mercy from Allah, not a deficiency. Prayer is a daily obligation that occurs five times a day; making up days or weeks of missed prayers would impose immense hardship. Fasting, being annual, is made up later when it is feasible.
If purity is restored during the time of a prayer (e.g., the period ends before maghrib has expired), that prayer becomes obligatory once ghusl is performed. The Shafiʿi and Hanbali schools also require making up the previous combinable prayer if there is enough time to perform ghusl and pray it (e.g., if purity returns before maghrib expires, ʿaṣr also becomes obligatory in these schools).
This summary draws on the established positions of the four Sunni madhhabs. For your specific situation, consult a qualified scholar.