Tea and sympathy to you 
sometimes you just have to do what gets you through. You can always, always deal with the 'rod for your own back' later, once it really matters and/or you have more energy to deal with it. Beating yourself up makes you feel worse and parent worse. (Voice of experience on that point!)
I am a lone parent working full time with a 10 and 5 year old. They each have a 45 min daily limit on the tablet - 10yo uses his up every day, sometimes entirely before school, other times split across (pretty much never all after school). 5yo can go a week or more at a time without bothering to pick it up, but she does watch more TV.
They both need to connect with me every day somehow. I have found that really short bursts of this can work well - even 5 mins for the 5 year old. She likes it to be her imaginative games which is often really easy for me even though 'playing' doesn't come to me naturally - she likes me to be the grandma while she brings her babies in the buggy to visit, I have to dote on the numerous babies, help tuck them up for bed, and just engage quite reactively with her play conversation. Just five mins. Makes such a difference. I can carve out five mins undivided attention for her at least once a day, and then I can also play along a bit whilst also sorting packed lunches etc, or I can get her involved in helping me cook the dinner.
The 10 year old gets his 1-1 time with me after she's in bed. A board game or vegging on the sofa together watching TV. It means sometimes there's a tricky balance of enough sleep and enough connection, but it levels out over a week or two as best we can.
Honestly in your shoes (and I really have been there) the only change I would start with is finding five minutes in the morning where you really focus on her. I know how hard that is especially with smaller children needing you too. And I also think you can put limits around it eg I refuse to do most things for my children (barring real needs, and excluding cuddles in bed) before I've had my coffee! But if you give her your attention proactively she really will start to back off as her needs are better met, i think. Don't choose activities which require significant prep or tidying up. Dont focus on the screen time or the overwhelm. Just find her five minutes.
Final thing to add is that this week we've all been cooking dinner together when we get home. I've avoided that approach for ages as it feels knackering and slow, but it's been SO much nicer than me in the kitchen while they sit with screens. I'm sticking with it, although I definitely need to collect a few recipes which are quick, easy, and have clear jobs for both kids. This might not be where you're at yet though - now my youngest is a bit older it really has opened up more possibilities for how our family life functions.
I think you have to balance recognising for yourself that it's impossible whilst making a good show of enjoying life for your children. I try to think about what childhood I want them to remember. They'll never have me picking up at 3pm and going to the park then coming home to paint etc as I might wish, but they can remember us being a reasonably cheerful team together and playing a little bit.