Wow, whoever installed that subpanel was just trying to get paid. Did not do you any favors at all.
It would have added less than $20 to that job to give you a 20-space and a 70A feed, that you could’ve done something with. As it is, the panel is “maxed out”, even overloaded, with washer/dryer going at once.
I don’t understand why you have two washer/dryers. If that is a separate tenancy, there are a moose-pile of other issues that should be considered prior to any panel upgrade.
Since you’re game to add a panel...
Let’s upgrade the panel you already have. This is easy-peasy, since this is surface-mount stuff in unfinished areas of the home.
First, move that junk out of the mandatory 30x36” working space and it never returns. Now, I bet that panel is connected to the main panel through a short EMT “nipple”. That’s the way to do it, but you’ll want larger pipe - like 1” trade size (about 1-1/8” size if EMT) or even 1-1/2” if the holes are present and will line up.
So shut off that 40A breaker, pull the branch circuits entirely out of the existing panel and carefully tuck them out of the way. Undo that EMT nipple, un-fish the #8?#10? wires, and remove that panel from the wall.
Now get yourself a 30-space or 40-space main-lug panel. The breakers currently in your Eaton panel are $20 to replace, so you could go Eaton BR to preserve that “investment”. Other than that, I’m a big Siemens fan, both because they’re the same breakers in your main panel so you can just move them over... and because Siemens makes a dirt cheap generator interlock should you ever have the fancy. Make sure it comes with/buy accessory ground bars; can’t use neutral for ground in a subpanel.
Use a big conduit “nipple” between the panels (and if the holes line up, several additional little ones are always nice). Use metal as that automagically takes care of grounding.
Wire the new panel
With the new panel mounted, reconnect all the existing wires and cables exactly as before, and get your upstairs laundry room powered back up. Everything is the same except now you have 22/32 more spaces than you had before.
I created this as a “break point” - you are back up to full functionality (as was before) and under no time pressure to do what follows. Better, the old work is a template for how to do the new work.
Now, look at the 40A breaker and the #8? #10? Wire going over to the subpanel. Replace the 40A with a 100A or 125A breaker, but leave it turned off.
Now, one wire at a time, replace the 3 existing THHN/XHHW wires with
- If 100A, #3 Cu or #1 Al wire, THHN or XHHW.
- If 125A, #2 Cu or #0 Al.
These wires typically come black. Use tape to re-mark the neutral wire with white tape.
Once all wires are replaced, you can power up that breaker. Now we have plenty of room in the subpanel, both in spaces and amps, and can rearrange as needed.
Another natural breakpoint. From here on out, you can move circuits around one at a time.
If you have a mind to “generator” in the future, move circuits you’d want on generator to the new panel, but reserve the top 4 spaces for the generator interlock. Also leave enough slack in the #3-#0 cable that it can reach a branch circuit breaker in the top 2 spaces, instead of the lugs.
Now, about that kitchen...
From the data you clipped off the instruction manual, it basically says “See nameplate” which you didn’t show. But we can make an educated guess.
First, a PSA for running 1” conduit between panel and the kitchen range/oven area, as a gift to your future self / future other homeowner. This lets you run whatever the future needs will be, up to 3 circuits. Further, since wires are allowed to run hotter in conduit, you can get more ampacity from smaller wires - for instance #8 is good for 50A in conduit, so a #8 run handles any future range/oven.
Code requires at least about 7 circuits for a kitchen these days. I always laugh when people build ADUs (in-law apartments) and they fit a 12-space panel.
However, you’d be doing yourself and the chef a favor if you also
Put the refrigerator on a dedicated circuit, or at least one that is not GFCI protected. The goal here is to prevent both a) nuisance trips (fridges trip GFCIs rather quite a lot, and needlessly since they are well grounded and shielded)... and b) trips caused by other things, which would result in spoiled food. That can happen if the resetter is unaware that thing also powers the fridge.
Have more than the statutory minimum two kitchen countertop circuits, so the chef can run more than 2 heat appliances at once, and doesn’t have to guess which plugs are on which circuits. (Heat appliances, typically 1500W, hog the entire circuit).