If you haven't done any exercise in 10 years, >5 miles for your first ride is great. So is doing it at ~10 mph.
Yes, I think you should build up your mileage with evening (before sundown, hurrah for Daylight Saving Time) and weekends. In fact, you should probably test-ride your commute on a weekend before doing it for a workday. Is there a good spot for you to have a rest and snack near your work site? If so, you could do your "morning commute" to that place, chill for an hour (having a drink and a snack), then do your "evening commute."
You didn't mention what kind of bike you're using. Everything will be easier if you have proper gearing (and the bike fits you). If this is the case, I think you'll be pleasantly surprised how much your endurance will build in the next month or so.
Hang in there, and good luck!
Edited to add: Just noticed you have a Fuji Absolute. That will be a fine commute bike.