I think OP is assuming tonight will be as bad as last night and thinking about calling in sick tomorrow.
First of all OP, sounds like you must have lucked out with a good sleeper if this is the first time you have faced being so sleep deprived you felt sick at work

I can't tell you the number of days I felt sick from sleep loss at work this year ... It seemed like months actually! Good times, good times.
The only time I ever called in sick was at the very beginning after I returned to work and the baby hit his 4 month sleep regression. (After this experience I tell every pregnant lady I know to take 6 months off to make sure they get past the sleep regression.)
But it sounds like you have not yet built up the ability to function at work as a zombie, like I have. So go ahead and call in sick tomorrow if you have another rough night! There is no reason to suffer.
The only possible problem I can see is using up your sick days when you might really need them later. My baby hit a string of nasty colds right around 11 months and I burned through my leave really fast.