She only trains from 4 a.m. to 8 p.m. every day. Could it be that you are approaching her during her "off-hours"?
You can try "getstage TrainingMarksman" (no quotes) in the console to be sure the quest completed. If it returns a value less than 100, try "setstage TrainingMarksman 100" to complete it properly. If it returns a value of 200, that means Alawen already died at some point (maybe you resurrected her via console?) and thus the quest cannot be completed. In that case, you'll need to find or make a plugin that adjusts that quest to make it completable in some way.
If all you did was set her to essential, that would not have affected her training in any way. If you edited the training quest at all, that could permanently affect your saved game, but the console commands above will get around that for you.
Just a note, too, the Unofficial Oblivion Patch makes her essential until you have completed the quests. You shouldn't be without the UOP.