Wife got a new 16gb phone, replacing her 32

[mainstreetmark]

She got a iPhonr 5S on friday (of all days), and immediately the new phone used up all it's space trying to restore. She has 0 bytes free.

So...
- I upgraded her iCloud storage plan again to 50gb
- I enabled the various iCloud/iPhoto settings
- I plugged it into her old laptop, but it thinks it's imported everything, though the pictures end last February on the laptop.
- I tried upgrading to iOS9 via the laptop, but even that requires 650mb
- The lock screen says its still restoring from backup, though without space, i doubt it'll ever finish

I consider myself an expert on all things Apple/iOS, but I have no idea how to proceed. All my normal attempts are failing.

They have 10,000 programmers there, all of whom eat the dog food they make. Why am I struggling with this? What do I do?

Frustrated.

(and forum searches haven't revealed any compelling how-to guides)

Edit:
- Cannot do a backup of the device to the computer, for lack of space.
- Laptop says the phone is over capacity by 104MB now, up from 20 a while ago, so it's doing something?

 

[mainstreetmark]

I restored the thing from factory settings. iCloud filled in the blanks with data. Lost all installed applications, and photos for the last 9 months. Thanks Obama.