Totally I didn't go to toddler groups with either of my DC because I was unable to walk when DS1 was little due to severe SPD in pregnancy with DS2 that never went away. My kids are 18&20 now, it never harmed them and tbh they don't know that we didn't go!
I think the important thing is to try to get on top of your housework. It sounds like it being in a mess is getting you down. You don't mention how old your toddler is, but when DS2 was little he loved cleaning. He would follow me around with some baby wipes, cleaning the radiators and skirting boards. He was quite happy to put things away if we made it in to a game.
Have you got a kitchen timer, or a stop watch/alarm on your phone? Set it for 30 mins and set to in one room. You'll be amazed as to how much you can get done. Maybe you could do that tomorrow with your gilrls, and they could help? Make it into a game and if the weather is dry and go to the park for a runabout after?
Dinner is a difficult one because you have to pick your DH up, but if your DD1 has school dinners can you give your toddler what you and DH had the night before and give them a light tea before you pick DH up? They could always have some toast and milk before bed if they are hungry.
Sleep deprivation is awful. I don't sleep well due to pain. I quite often have to go back to bed in the afternoon or not get up before 10am, or I just can't function. The more I just try to push on through the worse I feel. I completely understand why you go back to bed.
BTW you're not useless, you're doing a really hard job. Looking after children, on your own, 7 days a week is hard. Don't be so tough on yourself. 