Looks like you did a n LVM install of Mint, sort of. Did you mean to do that? Do you know what LVM is? You basically have overwritten your windows with your attempt and have not come close to completing the install of Mint. Do you remember which installation type option you selected? It should have been the manual option which I believe Ubuntu/Mint calls "Something Else". The LVM install is defintely not the default so I'm not sure why you chose it.

Your windows was an EFI install, standard with pre-installed windows 8/10 and that's all you have left is the EFI partition. You don't have any Mint files showing in the EFI partition so it can't boot. You also don't show the standard files shown on the Linux partition (sda5). There is also no windows system partition.

The only sign of Grub is on the flash drive installer. I don't know what instructions you used but the site below gives pretty explicit instructions on dual booting Mint/windows 8/10 using UEFI. Mint is a pretty popular system and you should be able to find numerous detailed instructions on different sites if you don't like the one below. I've never used LVM so can't help with that. If you had personal data on the machine with windows, I hope you have a backup.
http://www.tecmint.com/install-linux...oot-uefi-mode/

The link below is to the Mint site with explanations of various installation types.

https://community.linuxmint.com/tutorial/view/2191

@yancek Thank you for the response. I was actually following similar instructions to what you linked when I did this last night. I had issues disabling the secure boot on my PC and couldn't get anything to boot for a while. Then the Mint boot worked when I switched secure boot back on. In Mint I got to the screen attached but it did not have the "Install Linux Mint alongside Windows Boot Manager" option, it just had "Erase disk and install Linux Mint" and the "Something else" option. I clicked the "Something else" but then the continue button was greyed out so I couldn't click it. It was at this point I quit the install and went back and tried to boot into Windows again. I am unsure how the LVM install happened. Fortunately this a new computer so the only things on it were games from Steam and no personal files. I'm not worried about file loss. I downloaded a Windows 10 recovery disk and I'm going to try to boot from that/reinstall Windows on the machine and then I will follow the instructions for the Dual Boot again. Thanks so much for your help.

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In fact these are the exact instructions I was following. It was after this step
"Close Disk Management utility, place Linux Mint DVD or USB bootable image in the appropriate drive and reboot the computer in order to start with Linux Mint 18 installation.

In case you’re booting Linux Mint for installation from a USB dive in UEFI mode make sure you’ve created the bootable USB stick using a utility such as Rufus, which is UEFI compatible, otherwise your USB bootable drive won’t boot." that things went south.

How do I make sure Linux is installing on the correct partition? Would it be better for me to use a different Linux distro for dual booting? I'm not tied to Mint, just heard it was a good system for beginners.

Ok I've found the main issue. When I turn off secure boot, my Windows Boot Manager disappears and I can't boot Windows. Is there any Linux Distro I can use that doesn't require me to turn off Secure Boot (UEFI)?

Broke indeed.

Last 2 times I used it the distro wouldn't boot.

The last member I helped about 2 weeks ago had success with Rufus on his Windows machine.

If you don't see the "Install Alongside" option it is probably because you don't have unallocated space on which to install or you have left fastboot/hibernation on in windows 10. Mint or any Linux will not mount a hibernated ntfs. If you don't know what LVM is, you don't need it. You can clearly see that option in your image so make sure you don't select it. I don't use LVM but my understanding is that with the Ubuntu installer, if you select it it overwrites the disk and the explanation is not really clear on that.

There should not be any partition but rather unallocated space. You can create a partition in windows for Mint but it's kind of pointless as you will need to create a filesystem on that partition for it to be usable and you can't do that in windows. Just use unallocated space and create the partition/filesystem during your install.

If you want to try Ubuntu (Mint's daddy) you should be able to at least determine if you are booting UEFI. Documentation is at their site at the link below. I'm sure there are other OS's but I don't use UEFI so am not familiar with them.

https://help.ubuntu.com/community/UEFI

Also, looking at your boot repair output again, there are no Mint files in the EFI partition where there are windows files. If you reinstall windows again and use UEFI then when you install Mint, you need to also install Mint UEFI. I don't use Mint so I'm not sure how you would determine that on boot. Make sure the settings in the BIOS are UEFI I guess.