Here's the deal with tv: my deal, anyway.
When tv becomes a problem, it goes away, period. When we have fussing over turning off the tv, then we might decide that there's no tv for the rest of the day. Or if we were done for the day, then there's no tv tomorrow.
Simple.
As for what YOU are going to do: sorry, but this is one of those big girl panties moments: when my son is not allowed to watch tv, WE don't watch tv. Again, very simple. I'm an adult. I can live without tv for a day or so. I only have the tv on in the evenings after he's started his evening bedtime time. Occasionally I'll put something on for both of us to watch while I fold clothes or just for fun, but only if he's got tv time coming.
What I hear from your post is that you are feeling challenged with keeping your daughter busy. TV shouldn't be a replacement for engaging activity. Find out what you can offer her during down times. Can you sit at the table and nurse while she plays with playdough? Set a board game or puzzle on a table and let it stay there for a few days, work/play for five or ten minutes at a time when you can, together?
If it were my kid, I'd be getting together a bag full of fabric and paper scraps, found objects, tape, glue sticks, markers-- let her make paper dolls or create fun 'creations'. Let her bring out a doll and 'make clothes' with tape, buttons, a stapler... I'd say the more open-ended the offering, the more engaged she will be.
Get out a card table, cardboard boxes, blankets and pillows- instant fort.
In our house, I don't have a hard and fast rule about how much tv/media time our son gets. Some days we are so busy --- no time for tv! Other days, like when we are packing for a trip or if I have a lot of cooking to do, he'll get a bit more. From my best guess, two hours a day is our far-out maximum, just because we have so many other good things to do. (That two hours a day, again, might happen once a week or once every few weeks.) Our boy is seven and reading; each morning he receives a list from me of things he needs to get done before he gets any media or tv time. But attitude is the key determinant: if he gets huffy or refuses to turn off the tv, or if he nags me, he gets one warning and then that's it.
Oh, and when your daughter is fresh, send her to her room until she can find her 'friendly words'. Unless the problem is related to the tv, that can become an overused currency. Find out what she's unhappy about and help her express it more constructively. I often find myself asking children "what would you like to happen instead?" or "when this happens next time, what would you like he/she to do?" I mention this because while taking the tv away might shut her up for a moment, you aren't addressing what the underlying problem is.
Also, consider having your husband take the baby for an hour or so and do something fun with your Big Girl. Remember, I know you are all exhausted with a new baby in the house, and Alyssa needs attention. You say she's alone a lot. Take her to the park whenever you can. Get out. Personally, I really love the adventures my son and I have alone. Playmates are fun, but generally can be more work at times. Being outside, walking around on a forested or park path, going to the market and letting him pick out some fruit... little fun moments. I did childcare with some kids Alyssa's age when my son was four months old and I know how challenging it can be, believe me! It's a challenge to coordinate 'get kids from preschool/get them set up with an activity/get quiet to nurse baby to sleep/okay,now a snack for the kids/ clean up, oh, now baby is up... I get it. Prior planning helped a lot when it came to having things organized enough to grab them up (I did that in the evenings when husband was home, only took a few minutes).
Sorry this is so long, just felt that you needed more than how much tv time my son has....
ETA: I should add that, when the school year starts, we have tv ONLY when all our other stuff is done and never, ever before school. That's my only hard and firm rule on that one.