Stack height on a normal threadless headset (the type the Monocog uses) is right around an inch (+/-), so if you don't know the stack height of you headset, you could figure roughly 25mm+/-, plus the head tune length, plus stem and spacer(s) for a rough estimate.
So, in your case:
head tube: 110mm
headset stack height: 25mm
spacers: 30mm
stem: 45mm (that's what many I've used seem to hover around)
That puts you into the 210mm range or right around 8.25" as a rough estimate.
To be accurate, you would have to know the stack height of your specific headset and stem height along with the head tube length and spacer height. Then subtract 2mm to 3mm so there is room to preload the headset bearings using the top cap against the stem.
A 7-7/8" steerer could work with the 110mm headtube, but you would just have to adjust the spacers to fit and it would likely be less than 30mm of spacers depending on the height of the stem (clamp).
If you didn't use any spacers, you could probably get away with as little as 7"+/- of steerer tube with a 110mm headtube and standard threadless headset using no spacers, but you'd have to measure your stem to be sure.
