netstat reveals that there is still a socket open on the client

Which state is it in? Note that TCP requires participation of both parties to close a connection (see also: TCP TARPIT), so even if a socket-owning process died, the kernel still has to maintain some state about the recently passed connection.

the second attempt results in a message on the client that the server is not responding

This sounds more like a server bug, what does netstat say there? Capturing the exchanges (both successful and unsuccessful) with Wireshark might prove enlightening.