Take a deep breath. Stop stressing - it's not helping you or your DD2's. If you stress out in advance, it will be a nightmare!
If the youngest one is a rubbish sleeper whether you are home or not, just take each day as it suits you. When she is tired she will sleep. You will probably find she will sleep better when she has her big sister to play with and a lot of new things to see and do when she's awake.
DD2 may not be keen on the pushchair, but she is only 9 months, put her in it with some authority and get on with your day. She will accept this is what is going to happen when you do it, if you take her out every time she whinges, she will whinge.
Focus on activities that DD1 will enjoy - DD2 will just enjoy being in different environments...
Beach/Woods/Playpark/Softplay/Outdoor Pools/Farmyards/Library/
Friends houses for picnics & play.
Have some fun days at home:
Paddling pool: (if you don't have one), lots of places selling really big ones cheaply this year.
Garden Sprinkler: Lots of fun to be had there and it waters the garden at the same time - bonus!
Paint: Finger paint (hand, foot, whole body), makes a right mess - but a session under the sprinkler after sorts it out.
Sandpit/box/stand: Hours of fun
You will be fine if you 'get into the gooove' and stop thinking about it being a nightmare.
Alternatively, employ a nanny and go to work