Port 113 is for identd. DCC Chat and other DCC commands use the port(s) you have set up under the Ports button in mIRC's options.
I don't know if this will help you, but I've had issues with my IP not being filled off and on with Roadrunner internet. When I changed my DNS Servers, it solved the problem. I just used Google's DNS Servers (188.8.131.52 and 184.108.40.206). You can set those up in your network configuration. It's worth a try at least.