Why can't I delete the second partition

The reason is that disk0 contains your actual drive. If you look there, you see that only 125.4 GB of space is allocated to the "Core Storage". Disk 1 is the logical volume (Macintosh HD) which resides in disk0s2 (also named Macintosh HD) so deleting disk1 won't be what you want to accomplish and the tool is preventing you from deleting that "logical device".

You can fix this with the cs subcommand of diskutil in terminal to remove the core storage and put things back on disk0s2 in a none-CoreStorage format.

For safety's sake, make a good backup before continuing and then revert the core storage volume:

diskutil cs revert 8F5A679C-06AB-44CC-BD46-33E327446D80

Now let the machine revert to none-CoreStorage layout. Once that's done, you can use the Disk Utility and or diskutil to resize disk0s2 to take up all the space. Then you would re-enable file vault if you want core storage going forward. This may take more time than wiping the drive and reinstalling the OS and restoring your backup - but you will learn more by using diskutil and not need to test your backup now if you've never restored it.

I personally don't see any benefit to using Core Storage unless it's for encryption, but you're free to do so. Also, there is an undocumented diskutil cs resizeStack operation that I'd be hesitant to rely upon, but if you don't care to lose the data, might be a one trick pony to get your resize done.

I'd just revert the core storage to normal storage and then grow your normal partition using Disk Utility from reading the various comments and your initial post closely.