Sorry, but you seem to have fallen for an Enlightenment slander (like the belief that educated people used to believe in a flat earth). I once wondered what were the numbers that they suggested, but they seem only to have discussed whether they had corporeal bodies (and hence the number would have been finite), or whether not (in which case the number could be infinite).

Rather disappointingly, they seemed to think that the number being infinite settled it, where I was hoping they would try to work out the cardinality: for example, if they could argue (no idea how, but presumably from Scripture) that the angels all had names, taken from a finite alphabet, then the number would be recursively enumerable, but they didn't even go that far.

Physicists have come up with a different answer:

According to Thomas Aquinas, it is impossible for two distinct causes to each be the immediate cause of one and the same thing. An angel is a good example of such a cause. Thus two angels cannot occupy the same space. This can be seen as an early statement of the Pauli exclusion principle. (The Pauli exclusion principle is a pillar of modern physics. It was first stated in the twentieth century, by Pauli.)

[...]

We have derived quantum gravity bounds on the number of angels that can dance on the tip of a needle as a function of the mass of the angels. The maximal number of angels -- 8.6766*10exp49 -- is achieved near the critical mass mcrit>1/kD �3.8807*10-34 kg, corresponding to the transition from the information-limited to the mass-limited regime. It is interesting to note that this is of the same order of magnitude as the Schewe bound.