That's the thingy - (lightning, not fire-something; and £39, not £25). But....OP....."....For example, the Sky Go app will not allow you to use the HDMI output to watch shows on the big screen." :/
I have a dongle thing with an Apple connection (Fire-something) at one end, and HDMI female at the other. Providing you have a HDMI socket on your tv and a HDMI cable, Robert is your mother's brother. I think Sainsbury's used to sell the Apple dongle. Like all things Apple, it's not cheap (£25?).