It's worth seeing if you can find people working in education consulting already. They will often be able to give you invaluable advice on which firms to try, the nature of the job, etc.
If your university has an alumni directory, that is a good place to start looking. Even if there isn't a directory, your lecturers may be able to direct you to course alumni who went into consulting (or may have consulting contacts themselves).
Get on LinkedIn if you're not already - it may take a little while to build up your network as the public sector is less keen on networking than the private, but you're bound to find some contacts in the business that you never knew you had.