Your post led me to believe that your Toshiba external hard disk wasn't experiencing mounting problems, as you claimed. It just couldn't be modified, though it won't show up after using the two software.
Your drive is formatted using the Windows file system NTFS, which a Mac cannot read or write to. It worked before because of your NTFS driver. I guess the hard shutdown may somehow break the NTFS driver, resulting in you no longer can modify the data on the drive.
To fix it, you can either use a trustworthy NTFS reader for Mac like iBoysoft NTFS for Mac to enable the write access or get the data off the drive and reformat it with a Mac-native file system.
Whichever you choose, the drive must be identified on Mac to be fixed. I suggest you unplug and replug the drive into your Mac to see if it shows up. If not, try removing the two programs you installed just before the drive stopped being recognized by Mac.
Once your Toshiba external hard drive appears on Disk Utility, do the following:
Option 1: Free download and install the NTFS for Mac driver
If you need to use the drive on Windows or prefer not to format your drive and erase all data on it, use iBoysft NTFS for Mac, it's the best you can find on the market.
Once installed, the NTFS driver will automatically detect your NTFS external hard drive and allow you to read and modify the drive.
Option 2: Data Recovery + Reformat
If you only need to use the drive on Mac and want to use your drive without running any software in the future, choose this option.
Step 1: Download and install the best Data Recovery software for Mac. iBoysoft Data Recovery for Mac can restore data from unmountable, inaccessible, or unmodified external hard drives, USB flash drives, SD cards, SSDs, etc., from almost any brand, such as Toshiba, WD, Seagate, Samsung, etc.
Step 2: Open the data recovery software. Select the external hard drive you can't modify and click "Search for Lost Data."
Step 3: Once the scan is complete, you can preview your data in categories: Graphics, Documents, Video, Audio, Email, and Others.
Step 4: Check the box before the files you want to recover and click Recover to save them on your Mac or another storage device. (Can't be on the Toshiba external hard drive!!)
Step 5: Reformat the Toshiba external hard drive in Disk Utility.
If the erase process fails with the error that reads "File system formatter failed," you can partition the drive instead. In Disk Utility, select your drive, then click the Partition button.