From now there is a way to get notifications about new chat request just by using skype. Here is requirements
I won't talk there about sevabot install, you can find everything there http://sevabot-skype-bot.readthedocs.org/en/latest/ubuntu.html
As for configuring Live Helper Chat sevabot extension workflow is the following
Also for testing purposes you can edit sevabot/modules/test.php
If you are afraid of sevabot installation complexity there is simple option to buy my prepared sevabot image snapshot on digitalocean. Read here for more information.
You will receive notification about new chat request with link to accept a chat. It will be enough just click link to accept and start a conversation with client.
Developing application takes a lot of time. You can support application by donating. There is no company behind this application and it takes away my free time. Every donation matters and does not matter how small it is!