Did you have an engagement letter or some document describing this engagement? I always use them even for the most basic engagement to, well, c my a. It also lays out expectations including the responsibility of the client to respond timely. As BHoffman noted, you seem to be the initiator of all correspondence even when no reply has been received from a prior contact.
It is very difficult to care more than the client, isn't it? Yet we often have that nagging feeling that somehow we will be blamed. You might want to check with your insurer for advice. I've had to do it twice in 20 years and received good advice and a measure of comfort.
Yes, it might be time to fire this client but be sure to document pretty thoroughly the sequence of events, your advice and a list of tasks yet to be done for which you will no longer be responsible. Send certified, return receipt. And don't look back. If you haven't been paid, think about how much cheaper emotionally it is to be relieved of this situation. A year from now, maybe sooner, you will be so glad.