How can I make up the prayers I missed due to sleep or forgetfulness over approximately two years, knowing that I do not remember their number or times? And should I make them up in the order I mentioned (Dhuhr, then Asr, then Maghrib, then Isha, then Fajr), or should I start with Fajr?

It is permissible to sleep before the prayer time begins, even if one expects to miss the prayer, provided one takes measures to wake up. You have erred in your negligence regarding making up missed prayers, as the majority of scholars obligate making them up immediately. Strive to estimate what you have missed and continue making them up until you are reasonably certain that you have fulfilled your obligation. Do not stop unless you encounter hardship. If you do not know the exact number of missed prayers, repeat them until you are certain that you have fulfilled your obligation. If you forgot a full day's prayers and do not know which specific prayer it was, then repeat a full day and night's prayers. If you do not know when you first started missing prayers, you may begin making them up starting from the Fajr (dawn) prayer, though it is preferable to start with Dhuhr (noon) prayer.