I wouldn't rule out a dodgy splitter. However, the most likely cause is the differing resolutions of the screens. Next most likely issues are the Sky box HDMI chip set or running the splitter without an external power supply.
Presuming you have a power supply for the splitter, start with a look at the supported resolutions of both screens. The Sky box will output HD at one of two resolutions: 1080i or 720p. You should choose 1080i because that's what the HD channels are broadcast as. Sky doesn't broadcast in 720p, so the box has to make 720p from the source signal. On the whole it does a poor job of this. Do they both support 1080i/50 or at least 720p/25? Refresh rate can be just as important as resolution. Dedicated PC monitors are rarely happy at anything less than 60Hz or with interlaced signals.
Next, try the box direct to each screen. I know you've already tried that, but this time you're looking for any onscreen graphic from the TV telling you the resolution coming from the box. This will tell you if the box output is changing to suit the screen. It would do that if the Sky setup menu is set to Auto rather than fixed to one or the other HD resolutions.
You can also think about cable length, though there won't be much you can do to test the short link cable with the longer cables unless you have a HDMI coupler.
If both screens do support 1080i/50, power down all. Connect up but only to one screen. Make sure the switch is powered too. Then switch on the Sky box and that one screen. Does it work? If so, repeat but swap the screens over. Finally try with both screens. With a single screen connected the switch should be passing through without issue.
If you find that your problem is the screens trying to run at different resolutions then it'll be cheaper to buy a new screen than try to fix that.
