As I mentioned in the comments, the mechanism described in the OP is used in the laser cooling of atoms, and the detailed derivations of the force can be found by googling (although they may be complicated by taking into account the Doppler shift, which is not necessary here). I therefore make only a few points that are missing in the reasoning presented in the OP:
- Directionality the photons come from a certain direction, say with momenta $\mathbf{p}_i=\hbar\mathbf{k}_i$, but are reemitted in arbitrary directions.
- The rate of photon absorption and reemission the time of interaction of an atom and a photon is hard to define, i.e., one cannot really determine the force produced by one collision. On the other hand, one can calculate the net change of the momentum produced by many photons absorbed and reemitted, given that one knows how many such processes happen per second - here the Einstein coefficients could be helpful to relate this rate to the field intensity.
- Averaging As I have already mentioned, the photons are emitted in random directions. The changes of momenta transversal to $\mathbf{p}_i$ will average out after many collisions, leaving us with a net change (and hence the net force) only along the direction of the incident light.