Linux Mint not recognising free space created by Windows 10 shrink

I'd suggest using LVM. To switch over, something like this:

1. Create a physical volume (pvcreate) on your existing /dev/sda2 (Windows partition)
2. Create a volume group consisting of just /dev/sda2 (vgcreate)
3. Create a logical volume to hold your / and /home (any reason you're splitting them? You don't need to.) (lvcreate)
4. mkfs the two new logical volumes (device: /dev/mapper/vgname-lvname)
5. If you're not already booted from a rescue disc, boot from one (you could use the Debian install disc, just switch to a terminal at the appropriate point, or alternatively a Ubuntu disc)
6. mount the new logical volumes (or just one, if you decide not to split). If you mount both, make sure to mount home inside the new root (e.g., as /new-root/ and /new-root/home) Mount your old rootfs somewhere, too.
7. use tar to copy the files over. Something like ( cd /old-root && tar --one-file-system -c . ) | ( cd /new-root && tar vx )
8. chroot into your new root filesystem, and:
   1. change /etc/fstab to give the proper root filesystem
   2. mount /proc and /sys and /boot (inside the chroot)
   3. update-initramfs -u
   4. update-grub
   5. if you're paranoid, look at the generated /boot/grub/grub.cfg to check that it picked up the new rootfs.
   6. exit the chroot
9. Finally, unmount the filesystems you mounted (starting with /new-root/{proc,sys,boot}), and reboot.
10. Confirm you've booted to the new root filesystem. E.g., check /proc/mounts. Confirm your data is present. (You do have backups too, right?) VERY IMPORTANT.
11. This is the point of no return. The next steps destroy your old partitions.
12. You can now pvcreate your old partitions, then add them to your volume group (vgextend)
13. You can now resize your / and /home logical volumes using lvextend followed by resize2fs (or whichever tool for your filesystem). This can be done online, with your system running.

Once you're on LVM, any future disk changes are much easier. You can use the LVM commands to do pretty much any disk change, with the system running even.

(Note: I haven't done this for a bit, I may have missed a step or two, but I'm pretty sure I haven't missed anything catastrophic. And of course, your original install & data is still around until after you've confirmed you're booted onto the new rootfs)