Is it possible to find the energy level of a hydrogen atom in this way?

This is similar to Bohr's model, except that you are not using the balance of force (centrifugal force cancelling the Coulomb attraction). I haven't looked at the calculation in detail, but it would make sense that plugging in experimental values for the transitions, you would recover the same values of ##r## as in the Bohr model.

As @DrClaude say, it is similar to the Bohr's procedure and confinement of an integer number of wavelengths on the circular trajectory of radius r. But, remember: this model is limited and based on classical concepts. It doesn't reflect the complete wave-like features of the electron confined by the nucleus potential energy.

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DrClaude said:

This is similar to Bohr's model, except that it does not use a force balance (centrifugal force canceling out the Coulomb attraction). I haven't looked at the calculations in detail, but it seems reasonable that if we plug in the experimental values for the transition, we can recover the same values as in the Bohr model.

PeterDonis said:

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I see