Instead of going diret to the ticket office, why not use this site http://www.2for1entry.co.uk/ ; and find the London Tower link. It's often on the right hand side.
The title is self explanatory, and it can be booked on the web.
Also, the Tower want to charge you £15 per adult, with a transaction fee (http://www.hrp.org.uk/webcode/content.asp?ID=34), but the site will do it for £14.50. For 2 adults you could save £15.50.

Or depending on what else you want to do that day, why not try /www.londonpass.com/. You can get your travel and loads of discounts or freebies, including a 50 minute Thames cruise.

Many thanks for the excellent tips.