For the price of the Lightning-to-HDMI adapter and an HDMI cable, you'd probably be better off buying an AppleTV, they are £85 from the Apple
refurb store. We use the Apple TV to watch iPlayer streamed from iPhones and iPads all the time. You can also stream some of the other OD TV services, though not all of them (4OD doesn't work) . Plus of course you get all the other benefits of the Apple TV, e.g. Netflix, iTunes streaming etc.
It's a pity there isn't a BBC iPlayer app for AppleTV, but we can live in hope!!