Op, I'm so sorry about all you've had to go through. Others have come up with wise advice for you.
But am going to focus on DSD thingy.
The most important thing to them is to feel wanted and involved, even if you don't really. I'm sure you act a part, anyway, so you are just going to fo it even better!! And I'm afraid children aren't always grateful.
But remember, it needn't take lots of money.
Here are some suggestions for activities which you can play with half your mind.
Paddling pool, with you there to look after DD. Expect groans to begin with but get them to help blowing it up, filling it. Never failing fun.
Toe nail varnish. Take them shopping, 50p brands from Superdrug, each chooses a colour. They can have all their own colour or mix and match. ( buy one extra in case one gets spilt). They do each other's. Dd can have her pinkies done.
Decorating digestive biscuits. The holes are perfect for silver balls. Writing icing.
Picnic in park. They help make sandwiches, carry ruck sacks. They choose picnic spot. Then play area with DD.
Card games. Old maid. Cheat. Poker and vint et un. Best one if all though you have to buy the set, pit, which is happy families with shouting. Get DH to run them.
Making a collage for DD's bedroom. Design it first with them, based on what you know DD likes. Then decide if it's going to be paper ( easier) or fabric. Newspaper, note paper, wrapping paper, tissue paper, ribbons, stickers. Explain you want something made by them for their half sister, as they are sisters.
After that, Making their own pictures for their room, to make it really theirs. Could be their names, pictures, collages if eg their pin ups etc. Provide each with a large sheet of stiff paper, glue, felt tips, access to printer and internet or magazines, glitter.
Obv cooking, esp with DD. Biscuits in fun shapes are great.
Making stuff as a secret with them for their dad. Lots of occasions to think about: Birthday, Christmas, Father's Day, any old day -IRS an unbirthday present. If they say " but I've already done/got one", say "ah, but he hasn't got one from all four of you. Each of you has a separate thing to give him. But there's strength in numbers...a card/cake/play dough statue ( get them to make the play dough) from all of you is also special".
Playing castles in the garden if you have one or park with DD. Rug over chairs/ bench becomes castle. One child becomes princess/ besieged king with DD. Another is evil giant/dragon, another is prince/ rescuing forces. Then swap roles and who looks after DD.
Next variation is that rug becomes stage for band, playing air guitar, dancing.
The point if most of these is that DD can get involved and because she's so much younger, although the activities are younger, they'll get involved as most children are prepared to play with and look after younger siblings in my view. And taking them away for a period, not the whole time, from technology allows them to bond.
The older stuff, line YouTube and card games, are prob more age appropriate for when DD is napping.
If you get engaged in the activities I think you might find that your tension lets up a bit and you can find yourself laughing, they will bond with DD. They'll talk to DH about what a great time they had.
If they're sporty and you've got a kids area near you, that's good too. Swimming, with you always there for DD, is also good.
Hope helpful.