This is not the answer, but it lets me show you a picture of what should work when booting from a Live USB and reinstalling Grub.
The nicely laid out instructions from 'How to Ubuntu' on how to repair Grub with a live USB (http://howtoubuntu.org/how-to-repair-restore-reinstall-grub-2-with-a-ubuntu-live-cd) are unfortunately incorrect if using an EFI system. Adding one extra line lets it work, but it took a lot of chasing down other possibilities before realising this.
As an extra bonus, it also didn't actually work for you even with the line added :(
Here's my Live USB grub fixing that did work:
In this case, the Ubuntu installation was on sda7 (for you it was sda5) and the EFI partition was on sda2 (for you it was sda1). Everything else would be the same.
(incidentally, the 'grub-probe: error ' is for sdb1 which is the Live USB so can be ignored)
Why it's throwing up errors for you, I unfortunately don't know. The main reason for suggesting this was just a quick check to make sure that boot-repair had done it's job correctly and then we could look at other reasons for why you were getting the black screen.
I've used the same instructions before and they worked fine, but it must have been for a machine without UEFI.
I assumed it wasn't working firstly because the Live USB wasn't in UEFI mode, so might not have permission to write to the EFI partition, which in turn meant finding out that the Asus 'One Time Boot Menu' was accessed by pressing ESC on bootup and that they don't have Legacy Boot Options and UEFI Boot Options listed separately (which is what I've seen on a Dell laptop & an HP laptop) and it seems you need to choose 'Legacy USB support' to be disabled to ensure that it's booting in UEFI mode for your laptop (and I assume for other Asus machines as well).
Unfortunately we ended up down a rabbit hole of other possibilities before I realised the How To Ubuntu instructions are not taking an EFI boot partition into account.
If the now adjusted instructions do work, the main thing it helps with is to confirm that Grub is installed correctly on your machine.
It's a bit disconcerting that it's throwing up errors when trying to install Grub, but it might be best to just trust that boot-repair would give you errors if it hadn't been able to do it's job.
So use boot-repair as you did before, assume that Grub is fine and either troubleshoot along the lines of brndn2k's suggestion of looking here:
My computer boots to a black screen, what options do I have to fix it?
or try reinstalling Ubuntu again.
I've distracted you enough with this line of enquiry :)