I promise you can get to 100 a day in a month no problem. I started doing 10 pushups 5x a day and now do 20x5 a day. The difference in the way I look today is crazy from just 6 months ago, and it's not just the weight coming off. My arms and chest are bigger and more defined - I really feel like I look like I did in my 20s....maybe 30s...at least from the nipples/elbows up.
Try this schedule:
1st thing in the morning - like, right out of bed
Right before starting your workday
Just before lunch
Just after lunch
Knock out the last set before 5
Start w/ 10 each time, and move up as you feel stronger. 12x, 15x 20x...and if you can't get it done that set or that day, so what? Try again tomorrow. I have an accountability partner I text with 5x a day and a great thing is giving each other grace for not achieving that day's reps for whatever reason - sick, busy, taking the weekend off to celebrate or travel, it doesn't really matter the reason. You're doing this for you and as long as you are committed, you can come back and make it happen tomorrow.