Firstly, I'm really sorry to hear that you have been so unwell - my best to you.
Re the phone situation; it's done now so no point in getting into huge arguments about it however, at 15, your DD is old enough to deal with consequences of damaging another person's property. Perhaps learning the lesson now will help her work on controlling her temper in future where the consequences could be even more costly.
I would say yes - she has to pay for replacing the phone. How you organise it is something to discuss. I would have a calm conversation with her along the lines of "I get that you were angry, and we all have times when we have reacted badly to situations, however, it's not okay to have broken your sister's phone. I'm not going to scream and shout about it - but you DO have to replace it" - then figure it out.
You say your younger DD is happy with a cheaper phone - is she really or is she just accepting it as a poor alternative?
If she would prefer the quality of phone she had, then either a) you buy a new one now and come to an agreement with her sister as to how she will repay the cost or b) if your older DD has the kind of phone that you would need to buy - she has to sacrifice it to her sister and she gets the cheaper phone while she saves up for a new phone for herself.
If your younger DD genuinely is happy with the cheaper phone, then by all means get it but I would say that your older DD has to repay the cost of it plus a bit extra as compensation to your younger DD. Whether that is handing over money now or paying it off in instalments over an agreed period is up to you but stick to it.
As an aside - does your house contents insurance cover phone breakage? Many of them do these days - if so, you could get a new phone for your younger DD and have your older DD pay the excess - which would be a reasonable compromise.
I would consider what are feasible options here - then present 2 or 3 choices to your older DD and she chooses one. It holds her responsible - she has consequences, but having her choose also makes her think about it more and potentially opens up a dialogue about why she feels her choice is appropriate etc...