What is the ruling on fasting the fast of David with the intention of making up for missed fasts, while having forgotten the number of days, and how does one make up for missed prayers?

If the fast was broken intentionally without a valid excuse, you must repent and make up the missed days. However, if there was a legitimate Shar'i (Islamic legal) excuse, then only making up the missed days is sufficient.

If you cannot precisely determine the number of days you missed, continue making up the fasts until you are reasonably certain that you have fulfilled your obligation. It is permissible to fast one day and break the fast the next.

It is obligatory to pay a fidyah (expiatory payment) for each day of missed fasting if the delay in making them up was intentional and without a valid excuse. The fidyah is 750 grams of rice or its equivalent in the staple food of the local community, and it should be given to the poor.

As for making up prayers, you must make up the prayers until you are reasonably certain that you have fulfilled your obligation. These should be performed immediately, as much as you are able. It is not sufficient to make up each missed prayer alongside its corresponding current prayer.