Hi OP,
Looking at the photos you've put on, it doesn't look really 'dirty' or horrific. It's just cluttered and messy. The only issue with when it's gets too messy, is it's hard to actually keep the areas clean as all the stuff is in the way. The floors won't be able to be mopped when they have stuff all over them.
In the living room photo, there's just clutter on the floor, kids shoes on the floor, also on the stairs, try having a designated place in the house for shoes, maybe at the entrance if there is room? If not, put a basket/bucket as close as you can. Kids shouldn't really be kicking shoes off and dumping them about, do you wear shoes around the house?
The stairs, once the items have been moved to where they should be, it will just need a little hoover.
40 minutes to do you kitchen when you feel shattered and lacking motivation is really good going.
I always think to myself, if I didn't have kids my house would be immaculate, but I have them and now luckily in a good routine with tidying and cleaning. Mainly, as it creates a good environment for us as a family, it's calming and homely, mess makes me very stressed and unhappy. Tidy home, tidy mind.
I think try to be kind to yourself, you're not feeling great and that can make everything feel impossible and overwhelming.
Do you work? If so, it's harder to get the time in to clean, I now only work 3 days a week so I spend 3 hours each day on 2 weekdays to put music on loud and just have a really good clean, other days I just try to put things in the places they should be and maybe have a wipe around or a Hoover and at the weekend do 2 hours of cleaning plus sorting clothes and ironing. Spread I through the week.
Small steps to start.
I know you don't want to get the kids involved, but regardless of age, nursery years up, kids can absolutely be expected the keep areas tidy and clean their toys up after playing. If not, I'd be threatening that pile of presents are going on the fire!!
P.S I completely understand not wanting anyone in your house, it's your safe space.