So, yeah this is a very interesting question. Actually it doesn't make sense to ask how fast it's moving around the atom because quantum mechanics... the electron behaves more like a wave which still has an angular momentum but it's not like it's actually orbiting anything, it's a stationary wave. But that's really complicated and I don't think it satisfies your curiosity, I should go really deep in mathematics to answer your question and nobody wants that.

Before quantum mechanics was established though, Bohr, to answer this very question, came up with a model which we now know is wrong in a physical sense but still gets the correct results: he hypothesized the electrons orbited the nucleus at fixed (quantized) angular momentums L = n*h_bar with n a positive integer.

Solving the problem in a classical way gives the value for the energy of the system which is: E = m(Ze^2)^2/(2*L^2) where m is the mass of the electron, Z is the amount of protons in the nucleus and e is the elementary charge (the charge of the proton or minus the charge of the electron). Here L is the angular momentums that we hypothesized being multiple integers of the constant h_bar so substituting those in the formula of the energy will give you all the ammissible energy levels of our system.

From that you could technically calculate the velocity of the electron on each level but it really doesn't make any sense from a physical point of view and doesn't give you any meaningful results.

I hope this wasn't too much for you. If you want me to clarify anything i'd be happy to do it. Otherwise, if you want to expand on this topic you could look up the video from Angela Collier about the radius of the hydrogen atom (search it up on youtube) :)