http://www.perldesk.com/
and
http://phplivesupport.com/
are the only ways to fix this issue. ICQ has been garbage for a long time. It will piss off webmasters when you stop offering ICQ support but it beats having threads like this. Perl Desk will issue ticket numbers and track responses to the tickets so neither party can claim that the other hasn't responded.
PHP Live Support is a live chat that runs on your server. It will save logs and works every time. If your client has a browser, they can use it.