Hi there!
i am a mom of a DS who just turned two and am at home with him most of the time. I agree, being alone and stuck in doors with a toddler can at times be mind numbing.
We also have 6am rises, I dont deal with the dark cold mornings at the moment but do have plenty days with bad weather that i cant go out.. although its true that they aren’t great at independent play yet at this age, they can start building their attention span.
if you are feeling the screen time is too much, then listen to your gut? It takes more work from your end, but you feel better about it in the end.
you can try create a morning routine every morning, so obviously breakfast and getting dressed fitted into it, but maybe first reading couple books in pjs, then she helps with b/fast and rinsing her bowl etc - they love to do things independently at this age! And then have like a rotation of activities (take one thing out at a time) - colouring in or painting together, puzzles (2-3 pieces), shape boards, blocks, you can even work a ‘yoga or pilates session in’, my boy does some fun oilates moves with me, I am sure youtube would have something for kids or just make it up and play! Or playing dress up with her doll, and bathing her etc. if she already has a kitchen, you can pretend take the doll to the shops for groceries and cook for her etc etc.
and then if you feel you need to or if the morning drags on, by 08:30 she can have 20/30 mins screen time before you go out.. also try to hide certain toys in a box and do a rotation every so often, they will get excited about the toy they forgot they owned!
To help expand her attention span, try keep her engaged for at least 10mins on one thing by keeping it intriguing or not letting her go off to the next.. she cant do much pretend play yet, but you can!
does she like books? We do a lot of that on rainy days, and i have a book shelf accessible to him so he can learn to put the books back and get a new one (obviously not doing it neatly yet), he also loves to play ‘tent’ which just means throwing a blanket over chairs and sitting inside, any old activity suddenly becomes very exciting
i have also recently started more playdough and the other day we made things and played with play dough characters for about 30mins - even-though she cant do pretend play yet, you can suggest plenty things, have play animals talk to each other, play together etc and she will learn from you..
i realise leading the play in this way constantly is draining in itself and if you are not feeling like you have energy to deal with it it is hard - the little sun exposure in winter doesnt help with mood either.. maybe check with a pharmacist if some vitamin B and D supplements could help you?