Betts
I am on my second job through Betts. They have been nothing but great to work with, I would highly recommend them. On the secretive thing, they have to be. Sometimes discretion is part of what they do, and they also get paid for placing you. If they tell you who the employer is why not bypass Betts and go straight to them?
Sometimes the process takes a lot of time. My first job through them took less than 24 hours, the second took almost six months from initial contact until the deal was done.
Good luck and say hi to Fiona.