Hi there
There is a book called "Contacts" published by Spotlight. It's published every year. It's available from larger bookshops and it's also available on-line, either from Spotlight or from a site like Amazon. You can also get second-hand copies from a site like Amazon from previous years - these tend to be fairly cheap and will probably be ok for what you need.
You can then ring round all the children's agents in Contacts based in the North West and see whether they are happy for you to send in your child's photo and CV and then whether they are happy to audition/interview your child with a view to taking them on. Be aware that some of these agents will be modelling agents rather than acting agents. Personally, I would never pay an up-front fee to an agent but that's a personal decision. If your child likes performing then I don't think it's worth joining a modelling agency. Be aware that just because it's listed in Contacts doesn't mean it's a good agent (that's why I think the "no upfront fees" rule is a useful one!). Also, just because you're taken on by an agent doesn't mean you will get any opportunities or auditions.
A note of warning though: opportunities to sing and dance in a professional capacity (and professional work is really all an agent will be interested in, because that's how they'll make money!) will be very rare outside London. Think about any professional shows you have seen locally and consider whether there were any children in them. Even touring shows often use children from London. Many auditions take place in London too.
Opportunities for pantomimes might be worth looking at, though I suspect your dd will be a bit young for professional ones - she might even be a bit young for amateur ones: it depends on the company's policy. I believe professional pantomime producers will normally look for dancers from local dance schools.
Opportunities for TV/Film may be available, and the North West isn't a bad place to be for this. If this is something you think your daughter might like to try for then it's worth looking for an agent - see above!

If she really loves performing on a stage then I think she is likely to find many, many more opportunities to do this in an amateur capacity and I think if I were in your shoes, this is what I would be looking out for. You don't need an agent for this because there is no reason for an agent to try to find amateur work for a client (no money in it!). One way to do this (apart from the obvious "asking around", going to the local library etc) is to look at local venues/theatres within a radius of 15 miles or so and look at a year's worth of "what's on". You can then track down the amateur companies/youth theatres who put on appropriate shows appearing at the venues and see whether your daughter could join them. Another useful avenue to explore (though your dd might be a bit young yet) is to look at website for professional producing theatres near you and see whether they have an education department. They often run workshops or youth groups.
Hope that's helpful!