Ebay tends to be really quite expensive for second hand kayaks, so it's best avoided in my experience. Mostly bought by people who don't know what they're buying.
I'm not sure what kind of kayaking your daughter is into, but boats can be really use-specific so you need to be aware of what you're buying and what it's going to be used for. If you don't know then it's really easy to buy the wrong one.
A new kayak is going to set you back between £600 and £800, although there are some 'club spec' general purpose kayaks around for about £450 new.
You're probably looking upwards of £250 for a decent, flat-hulled, second-hand modern kayak. It's probably not worth buying an old huge long pointy boat, even if it is cheap. Kayaks and especially kayaking technique has moved on considerably in the last ten years, and those old, pointy boats are pretty much well ouf of date now.
Also, I don't know if your daughter is a member of a club, or if she just fancies messing around in a kayak. If she isn't already, I can't recommend enough joining a local club. Most clubs have are fairly cheap to join, provide good instruction and development in a safe environment and often provide much of the kit required, to borrow whenever you like, as part of the membership fee. Joining a club really is the way forward.
Most importantly of all though, don't even think of buying a boat and letting her go out on the local canal, river or lake if she isn't experienced or supervised by someone who is. If you don't know what you're doing or are inexperienced it is very difficult to spot hazards, and ALL water is dangerous, moving or not. I can't emphasis that enough.
Having said all that, Brookbank in Manchester, Robin Hood Watersports in Leeds, Desperate Measures in Nottingham and Performance Kayaks in Weston Super Mare are all really good (other kayak shops are available) and will give you sound advice, and they won't sell you something that won't fit you, or your daughter's needs.
Kayaking is a great sport though, a little effort to learn the basics and it can change your life. Kayaking has taken me to some truly amazing places around the world and lead me to share some brilliant times with some fantastic people. It's a great community and we live in a country with some world-class rivers.
Sorry, turned into a long post.