Since our talk, we've spent the past nine nights together and have had sex. She's seeing someone else, but hasn't talked to him since we started sleeping together again. She won't admit it, but I can see more in her eyes than just sex. I would love to rekindle this relationship, as I do love her very much. All signs in your letter point to a second time around.
Many relationships only become stronger when renewed after a breakup.
As you're spending time together again, make sure you discuss what went wrong over a month ago, and ways that it could be handled more constructively the next time. For instance, you could agree to spend less time getting involved in particular work projects, and your girlfriend could agree to let you know when you start ignoring the relationship. Whatever you two work out is fine, as long as you resolve the last problem and don't get into a pattern of using breaking up as an easy solution. Unresolved issues always come back to haunt you. So, talk it out.