My dc's were both this way, maybe not as angry, but definitely with very little sleep. If they are not sleeping well this can get really bad all around, the only ways we got the kids to sleep: let them sleep on or next to us, driving in the car worked well for naps too. Ds would never let us transfer him from the car or stroller, dd does ok with this, but will stay up all night so she can be with me. I might try again soon to get her into her little toddler bed with the lightning mcqueen sheets.
Anyway you might focus on physical exertion to get the baby to sleep more, with summer coming on a parent-child swim class might be just the thing, you will still be holding her but she will weigh a *lot* less! Hopefully after that she will crash hard for a nap!
For that matter, tub-time can be fun too, I know you say she can't sit up, but you can get in the tub with her to just chill/preserve your back. Some colorful cups are good if she's grasping stuff already (bet she is).
Also, I have to second the back-pack for kids, it worked wonders for us, not just for hiking. We used it to cook, clean, whatever needed to get done. Ds and dd both seemed much more content to 'oversee' than stay in a playpen.
Welcome!
