My DD is 2.5 and my baby is 5 months so similar ages. DD is also really bad at independant play. Particulary if that idependant play means I am doing stuff with the baby.

My 5 month old only has 2-3 day sleeps (mostly only for 40 mins). Occasionally I get the odd 1.5 hour sleep out of him. Most mornings of the week I have something on with the toddler so baby sleeps in car or pram. If I am having a bad day he just goes into a sling and I wear him while he sleeps - DD doesn't really like that very much though and tends to get a bit cranky. Whenever the baby is sleeping I make a big thing of doing stuff with her and letting her now it is special time cause the baby is asleep. That way she is happier for me to spend the time putting him to sleep.

I don't do housework during the day and try to find games that I can involve both kids in. Reading books, playing outside, paddle pool, blocks, etc. I also involve her in everything to do with baby. So its her job to get the clean nappy, spew rag, toy. Somedays it works really well other days nothing is in sync and its a nightmare

Sorry not really any great tips there. I am sure it will get easier as the baby gets older and more interactive and able to play with the toddler. That way they can play together (fingers crossed).