A C172 pilot failed to file a complete flight plan after diverting to an alternate airport on an instrument training flight.

Narrative

I was receiving recurrent instrument training from my CFII. We diverted to a separate airport due to a runway closure. We had not filed a flight plan from this airport to our home base. We asked ground control if we could file a flight plan with him; to save time. Ground control told us to standby for a moment; and then immediately came back with our clearance; without me telling him any of our flight specific information. Upon receiving said clearance; I quickly ran over my blank flight plan form; telling him the information I did not give to him; such as our alternate; souls on board; etc. However; I did not mention our estimated time enroute. While the weather was MVFR; had I done the same thing on a flight with IFR; and we lost COMMs; this omission could have presented a logistical challenge; as ATC and my CFII and I would have not known when I would have commenced the approach from the IAF in a no radio situation. For a corrective action; we should have gone into the FBO and filled with FSS; and not been a rush to get home and 'file' our clearance in a non-standard way.