Hi, we live at the other end of London and so I know very little about this school - except that it is very academic and very difficult to get into. Your daughter will need to be highly academic and very confident socially to thrive there. That said, it does have a reputation as a great school.
You should definitely consider alternative options such as Latymer Upper (co-ed, also located in Hammersmith) and Goldolphin & Latymer.
Some North London schools form the North London consortium, which is split into two groups of schools. You can choose to take an exam for either or both groups and apply to as many schools as you like in each group. The exams take place on two successive Fridays in early January. SPGS (common acronym for the school you are looking at) sets its own exam, also taken in early January. In the case of SPGS, this is preceded by a computerized aptitude test and all schools will interview applicants before offering a place. Most if not all schools will have sample exam papers on the websites (look under applications) - if SPGS doesn't, look at the website of Channing School in North London.
These exams are similar to, but not the same thing as 11+ exams, which are set by the few remaining selective state schools in counties such as Buckinghamshire, Hertsfordshire, Essex and Kent. These are at least as competitive as the entrance exams for independent aka public schools because they may lead to a free place at a very good, highly selective school.
You might like to look at the Good Schools Guide (available for purchase as hard copy or as an online version) to get a good overview of different schools in this country.
For more information on the 11+ exam, as well as on independent schools such as SPGS you may like to look at this website www.elevenplusexams.co.uk/forum/11plus/viewforum.php?f=31
as well.
Hope this helps a little - changed your mind yet about moving here after reading my post 
?