I’ve had to memorize several different pieces in short amounts of time before, and I’ve tried different things.
Definitely do not wait to start on the second one until you finished the first one. If you do that, you have to memorize each piece in only four days. If you work on all of them every day, then you have 20 days for each piece.
Do start by playing all of them at least once a day. If you can have two separate practice sessions a day where you play through all five in each session, that’s best. So if you really want to do this ASAP, get up earlier and play all five first thing in the morning, twice if you have time, then also play all five once or twice again before you go to bed.
After a few days of that, you should start to have parts of each piece memorized and other parts that aren’t sticking as well. Keep playing each piece all the way through and then go back and repeat the parts that aren’t sticking a few more times.
Also make notes in pencil on the pages about things that you keep forgetting or that trip you up. Actually write those notes - the process of writing helps you remember because it has to be processed by your brain differently.
If you have to play 100% without the sheet music after 20 days, then at least five days beforehand or as soon as you think you can, start playing through all five without the music. When you get stuck, look at the music and then close it again and play what you got stuck on. If you have time, go back a second time and try to get through the part you forgot.
Absolutely do not miss a day of practice and absolutely play every piece at least once a day. If you can’t practice morning and night all days, that’s ok, just make sure you get at least one run through of everything every day.