Is it possible to combine making up missed prayers with performing supererogatory prayers, the prayer of need, and the prayer of Istikhara, or must one finish making up the missed prayers first?

All praise is due to Allah, who guided the questioner to repentance. She must make up the missed prayers immediately. Taking the initiative is more important than engaging in supererogatory acts. If the missed prayers are numerous, a Muslim should make up two days' worth of prayers or more per day. If performing supererogatory prayers does not conflict with the required amount of make-up prayers, then there is no harm in her praying as many supererogatory prayers as she wishes. However, if the supererogatory prayers conflict with the make-up prayers, then it is obligatory to focus on the make-up prayers because they are a fard (obligatory duty).