It is sensible for your mum to stay home and you have the weekend with DH for him to recover. And regardless of illness sometimes you must have days/weeks when you might need a break and you should be allowed to do this without fear of upsetting your mum.
Is your mum capable around the home could she get to the shops or cook for herself independently? If she’s capable of driving or going out independently by foot or public transport can she think of things she can do to occupy herself? What options does she have to get out the house? Can she walk to a shop to buy a treat with her lunch, go to a coffee shop, museum etc? What are her interests? Does she make things?
Perhaps you can use this as an opportunity to switch things up a bit in the new year and help your mum engage in some activities independently. It sounds like you’ve tried but it might be worth trying again. You and your sister obviously live your mum but it’s a lot of pressure on you both to ‘cover’ her week.
Age UK do a lot of activities from exercise classes to social gatherings, what about WI or does she have neighbours she could invite for coffee? Is she involved in a church? It will be hard at first but she might really enjoy engaging with others outside of family.
Another thought, is she isolated where she lives ? Would a move to a retirement village be something she’d consider? Finances might dictate but there a different types that might suit. They are often independent living but have a community and some communal spaces and activities on site. There are lots around. My parents had a new lease of life when they moved to one. Since dad died mum has made new friendships and joined new activities. It’s fab as there is so much going on on the doorstep- there’s often an option of 2-3 activities each day, from a coffee morning to scrabble, knit & natter, exercise classes, talks, quizzes, music & entertainment nights, occasional theatre trips & excursions. All very tasteful. There’s a communal lounge & restaurant on site. It’s a thriving community where you can engage as much or as little as you like. It gives me peace to know she is getting on with things and has options when I’m not around.
Anyway I hope your DH gets well soon, you avoid the flu & you can find a way to support your mum that is a balance for you all. It’s not easy.