Wow. You rock, that's about all I can say. My kids will both be 3 and 1 in about a month, so I do understand the baby with the 2 year old, but you have two babies...yipes! I don't have a lot of advice for you except that I can only imagine how hard it is and you will get through it...eventually! ;)
I also think letting her watch a little Dora each morning is not a big deal. Of course you want her to be more active, but honestly, Dora is pretty good and she can learn some stuff! In fact most of those Noggin shows are awesome and so is Sesame Street. I usually DVR them and then I can show one or two and turn it off so it doesn't just keep going and going. Give yourself a break and nurse in peace (which is awesome by the way that you're nursing two babies!).
Also try taking walks around the neighborhood or to the park. THat way you're active and it's okay if she's running around like a crazy person. My daughter loved to push her baby in the stroller while I pushed my baby...very cute. I also did the same thing as Victoria and did a nursing bag. I bought a cheap bag at Michael's and some puffy paints and she got to decorate it. Then I filled it with books and things she could do all by herself (stickers, magnadoodle, etc.) and only brought it out while I fed the baby. It worked pretty well.
The afternoon is the hardest regardless of babies or not. It's just a restless time. Maybe you could save Dora for the afternoon or maybe you could plan a late afternoon playdate with a friend in the same boat. I have a friend and both our husbands have a tendency to work late so every week or two we have late afternoon play time where we hang out until 5-5:30 and then come home and fix dinner. My daughter loves to help with the baby, putting away groceries and cooking/baking, so maybe get her involved ina little of that too.
The last thing I want to say is if you want to go out (like you mentioned the library, etc) just keep doing it. Obviously if it's too stressful right now, don't, but it's good for your oldest to see that you have expectations of her and expect her to do the right thing (within reason of course, being that she's 2) and take advantage of the time when the babies are immobile. I know those doubles can be bulky and hard to maneuver, but go to places that it's okay for a little noise or where people expect busy toddlers. Barnes and Noble does story hour too and I've never been to one where all the kids are sitting and listening intently! ;) In fact, any mall is pefect for a 2 year old to get some exercise and you can push the stroller! Also, make sure you have a couple good mom friends that you can rely on...that will make a world of difference! Hang in there, sounds like you're doing great!