Ah! That is a common event. I have this, too.
For me, for a while, while working on inner heat concentration, I was generating inner cold for about two months! Along with inner cold, I would get extremely fatigued. It got so that within less than a minute, even in group mediation where there is some bolstering effect of a group even, I would get very cold, start shaking and want to go to bed.
I did go to sleep on most occasions, then stopped meditating for a week.
Then, I thought, "This is nuts! You have to get through this", so I sat and stayed awake, tested a statement in my head (I was going to go through the knowledges of suffering by stating, "I am misery, I am averse...", but as soon as I mentally thought, "I am afraid" the cold and fatigue stopped and heat came up!)
I am not saying that the cause of your sleepiness has the same underlyer or resolution as mine, I am just sharing that I understand the challenge and the sense of having had an worthless effect and a drowsy foe!
I think the general encouragement is to try again. In my case, taking a week off out of frustration (and some disgust) was useful. When I sat again, I was alert and determined, not beating myself up every day. It was useful to take a break and let intention re-build.
Good luck.
edit: brackets
edit 2:
more from the same monk as above and on drowsiness now. Also, why is it a problem that productive meditation occurs after a nap? That sounds reasonable; a person has some needs for rest.