Is it permissible to observe the fast of David with the intention of making up for missed fasts and prayers, given that their number has been forgotten? And how should one make up for numerous missed prayers?

You must make up for the days of the past Ramadan before the coming Ramadan, then make up for as many of the previous Ramadans as you can. If there is no time left, then fast Ramadan [current year's] and then continue making up [the missed fasts] in Shawwal. If continuous making up of fasts is difficult for you, it is permissible to fast them like the fasting of David, one day fasting and one day not.

If you were unaware of the impermissibility of delaying making up fasts until the next Ramadan, then there is no expiation (fidyah) upon you, only making up the fasts. As for the most recent Ramadan, its missed fasts must be made up before the coming Ramadan begins, otherwise, expiation (fidyah) is obligatory along with making up the fasts for each day, unless there is an excuse.

If you were aware of the impermissibility of delaying for past Ramadans, then along with making up those fasts, you must pay expiation (fidyah) for each day. The expiation is one mudd of wheat for a poor person (750 grams) or half a sa' of other grains like rice (1500 grams). The expiation is not repeated for repeated years.

To make up for missed prayers, you should busy yourself with making them up as long as it does not cause you hardship in your body or your finances. If you do not know the exact number of missed prayers, then you make them up until you are certain that your responsibility has been discharged. You should limit yourself to making up the obligatory prayers only, without the supererogatory or Sunnah prayers.