Pareto efficiency (optimality conditions) in simple New Keynesian model
The Lagrangian is also a function of the multipliers $(\lambda_t, \mu_t, \omega_t, \alpha_t)$ so you are missing their FOC's (these will give you the constraints as FOC's so if you were already taking them into account, omit my comment, please). Also, $-u_{H}=u_l$ since $l=1-H$ (I would re-write the second FOC, in terms of $u_l$ in order to construct the first "target" equation that has the MRS=MRT. Finally, there are a couple of useful facts that should help you. First, note that in your 4th FOC, you have a product equal to zero, so at least one of the elements is equal to zero. Try to convince yourself, that the only possibility (so that you don't run into a contradiction with the other FOC's) is for $\pi_t=0$ as required by your second-to-last target equation, and in turn implies $C_t=Y_t$ (from your first constraint). Second, using the second constraint you can replace the integral in your last FOC with $Y_t$.

This should help for further simplifying the 10 FOC's you should have, into the 6 target equations you want.