I know how dreadful sleep deprivation makes you feel, (I am there myself right now too) but I wonder if you keep laying her down for the full 3 hours, night after night, she will get it and understand that it is time to sleep?
My Mum and Dad said they used to have to do this with me, I was in a cot in their room and when they came to bed, they'd have to turn the light on to get ready for bed, I would wake up and want to play, but they lay me down again and again and ignored me! Initially I was upset about this, but I soon learnt they said that it was night-time and I shouldn't stand up and play, I think I was around 10 months at this time
I'm not saying you should do ignore DD, times have moved on! but she is old enough now to understand that it is no longer night-time play-time if you are consistent with her night after night, I think that's the key to it, is being consistent and not giving up, even though you may be shattered and completely bored with the whole thing of laying her down repeatedly
To an outsider, DD is having all her needs met, yet DH, DS and you are not so it's not balanced anymore, you have to take a little control back from her (for her best interest) and very kindly gently let her know that it is time to sleep for everyone in the house, including her!
It's really a tough call, at some point we start to have to withdraw a little from baby controlling the house, to us controlling them a little as they grow up, now if only I could take my own advice I'd be doing really well
But my DD is only 6 and half months so we are some way off yet of her really understanding why she is being denied something
You could try reading/googling Dr Jay Gordon for his advice on night-weaning - it might help you look at this as a habit and his night-weaning plan gives very kind advice on how to break a baby's habit of nursing all through the night, in a very kind way, lots of people use it on this forum and have good things to say about it
It seems like after 12months, babies understand that they aren't always going to get exactly what they want and this is not going to hurt them as long as it's done in a kind, consistent loving way
Anyway, hope the 45 minute nap helped! Hopefully you won't need to think about any of this any longer and can ignore everything I've written here!