Okay, so, I've got my Sky box in the living room, with the HDMI output on the Sky box going into a 2 way HDMI splitter, with one of the outputs going to the TV in that room, and 15 metre HDMI cable from the other ouput to a TV in my family room.

So, have added a HD TV in the kitchen, next door to the family room. Could buy a 3 way splitter to replace the two way one, and run another 20metre HDMI cable to the kitchen TV, but means lifting floorboards again, so was wandering, could I buy another 2 way splitter and have it in the family room, connected to the first splitter, and then run a short Hdmi cable to the kitchen?

Or does that give you a loss of quality?

Digital either works 100% or it doesn't. If it works there will be no loss of quality.

However no idea if you can get away with splitting an already split signal again.

There's no way of knowing in advance, some makes of splitter might work (especially seperately powered ones).

I'm using two splitters daisychained so it can work. I think the maximum is four levels deep, so 2 should be fine. I agree with the above - get a powered one.

sky only works with powered ones

Any HDMI splitter needs to be powered for it to work with two TVs at a time no non powered splitter can do this.

The question would be whether the Sky HDMI 5V output provides enough current for a particular splitter to function. I don't think you can make a general statement about splitters - it may even be that different Sky boxes have different current capabilities on the 5V, I have no knowledge on that point.

If I was to use component connection between the sky box and the TV beside it, would that give me a 1080p HD signal?

which would mean I could move the HDMI splitter through to the family room, using the existing 15m HDMI cable as a link, and hook up the family room and kitchen TVs to it?

If it is true component (requires 3x video and 2x audio leads) then it should do HD if Sky allow HD over component. They could restrict it to SD for copy protection reasons as it would bypass the copy protection used over HDMI for recording.

You'll get 1080i via component on a Sky Thomson box, you can source the audio from either analogue (phono)/digital (optical) outputs.