I would bill in 15 minute increments, as Julian has suggested...unless you advertise that you bill to the minute in which case you may want to revisit the logistics of this project with your client (so they're aware of the change in billing procedure)
It's not necessarily a mistake to have accepted a lower rate client when you're just starting out - many of us have...just be sure that you're using the experience to not only service your client but truly grow your business. This is a great way for you to learn more about your own expectations, boundaries, etc. so take full advantage of it!
