■ねむいねむい。

The countdown to fall asleep has already begun...

There were too many things to do in my office on a day like this.

Umm...

I'm not sure if this is the noise you mean. ;) "Umm" is usually hesitation, uncertainty. It's like 「えと。。。」 maybe? If you want a yawning noise, I'm not sure there's a good one ... so I usually just write "Yawn."

Since I slept only for 2 hours last night, my eyelids are coming down drooping now.

"Coming down" is understandable and not actually wrong, but "drooping" is more natural. :D

The countdown to fall asleep already has begun...

The other correction is also correct, but "has already begun" sounds suspenseful, like something is inevitably going to happen. "Has begun" is like a plainer announcement, with emphasis on the beginning itself.

I just made this correction as an option. You can pick whichever correction fits better. :D

There were too much many things to do in at my office on a day like this today.

"In my office" is OK, but "at my office" sounds more natural, and clearer, to me. ("In my office" could refer to the physical space of an office that belongs to you alone, while "at my office" refers to the whole place where you work.)

I changed "on a day like this" to "today," but that might not be what you mean. If you say "There were too many things [...] today" then the meaning is straightforward: You had too much work to do today. But if you want to say there are often days like today, when there's too much work to do, you could say instead, "There are too many things to do at my office on days like this" or "There is too much to do at my office on days like this."

Also, to be very very natural, you could say "at work" or "at the office" instead of "at my office." ;)

they're all wrinkled), so I'd get to sleep tonight.

Right now, this sentence is a little ambiguous. It could mean you'll only sleep because your eyeballs have become like umeboshis. ;) If you mean your umeboshi eyeballs indicate that you will sleep well, you could say something like "Now, my eyeballs are like umeboshis, so I'll definitely sleep tonight!"