If you believe that your girl is egg-bound, I suggest giving her an Epsom Salt bath since it helps with relaxing your hen and can pass the egg that way. If you're having trouble determining whether or not she is eggbound, if she does poop then she is NOT egg-bound.
As for the lethargicness and pale eyes, you can give her some electrolytes. If I find that my chickens aren't feeling that well, I put a couple of drops of VetRX in their water.