Do you want HD in the Kitchen too?
If no, then stay with coax it's inexpensive, and as Lurch says go for a good quality coax, no good reason why the sound is low, if you have another TV try on that and see if you get any change e.g. process of elimination
If yes
At 6m you could run an HDMI though a 2x splitter, the only issue is routing the cable as they can be large diameter wise and the radius on bends etc
The CAT6 solution is a more expensive choice (due to the sender/receiver costs), but likely easier to feed cables, since you only need a 7mm hole and you can dress these as the crimp tool and RJ45 are quite cheap
Lastly look at the HD wireless solutions, however with the