Best approach is to solve whole thing - start with equations describing all equilibria present in the solution (actually there are only two of them that matter), add mass/charge balances, then solve the system of the equations.
Not much different from what is done here:http://www.chembuddy.com/?left=pH-calculation&right=pH-salt-solution
Just Ka (dissociation of the HCl, not of the NH4+
!) is so high it can be assumed to be infinite which means we can ignore it. That leaves us with two equilibria - water autodissociation and ammonia reaction with water.
Don't think in terms of ammonia Ka and Kb - whole system is described by either Ka/Kw or Kb/Kw, using Ka/Kb/Kw system just adds an equation that is not independent (ie it doesn't give us any additional information, as it can be expressed by other two). After all, Ka*Kb=Kw, so knowing any two we can calculate the third.