I also think an agency is a good idea - if you were going to be around DC to interview candidates until you found the right one I'd say 6-8 weeks, but you'll have limited opportunities to meet nannies and, if I understand correctly, no time to see her interacting with the baby! You'll either luck out and find a gem or, more likely, feel like you have to settle on someone you're not 100% on because time is running out.
Sign on with an agency and you'll have the first leg of the hunt done for you (background check, references, etc.), access to temp nannies if interviewing takes a while and you both need to be back at work, and if you find a good agency they'll be able to help you figure out nanny taxes, benefits, duties, etc. - which is very helpful guidance for first time employers. Good luck and congratulations!