Does the TV not have a VGA/DVI plug?
They look like this:
http://www.pcworld.idg.com.au/article/346482/how_use_your_tv_monitor/
It might be labelled "PC".
Don't worry about the router and ethernet - I've never known anyone to plug in their PC to their TV via this method.
If you TV has a VGA (blue) or DVI (white) plug, you may simply need to unplug the cable from the back of your monitor and plug that into your TV.
Picture quality is kind of non question. It's likely your Monitor/PC can produce a signal better than the "HD" of your TV. You may need to scale the resolution of the PC up however.
Let us know how you get on.