I have the exact same laptop as you, the Latitude E7450, and had the same exact problem as you. I was very impressed with my own solution and I thought I would post it here:
1) Hit the WINDOWS key.
2) Type "Dell Touchpad" and hit ENTER (without quotes, into the Windows search box - my cursor lands there by default after hitting the WINDOWS key).
3) Use the TAB button to cycle through the application. The TAB button does work but it is very hard to see the outline of the buttons for which it cycles.
4) Hit TAB five times. The first click of TAB takes you to the minimize button, the second click to the close (X) button, the third click to the version information on the bottom left, the fourth click to the "Display Icon in System Tray", and the fifth and final click to an unmarked location which is presumably the touchpad icon toward the top of the black window.
5) Hit the DOWN ARROW key twice. The first depression will take you to the "Default" button, and the second depression will take you to the "Sensitivity" button.
6) Hit the RIGHT ARROW key once. The first depression will take you to the "Touchpad On/Off" option.
7) Hit the SPACEBAR to enable/disable the touchpad. This is not complete until you click "Save".
8) Hit the RIGHT ARROW key once. The first depression will take you to the "Tapping" button.
9) Hit the UP ARROW key once. The first depression will take you to the "Cancel" button.
10) Hit the LEFT ARROW key once. The first depression will take you to the "Save" button.
11) Hit the SPACEBAR key once. The first depression will save your settings.
12) Use the touchpad
13) (Optional) Click the "Tapping" button and "Enable touchpad by Zig Zag gesture". The zig zag motion must be wildly exaggerated to work. This will save you some trouble later if you find yourself with a disabled touchpad and no mouse.
If at any time you find yourself lost hit ALT+F4 and start over.
This took me about an hour of trial and error to resolve. I was determined.
The "Dell Pointing Devices" program, as found from Control Panel e.g. via "Dell Touchpad" shortcut, can't be used with those "Tab" etc. keys, contrary to normal Control Panel subprograms.
Perhaps this quite trivial use case slipped through when defining the use cases for testing?
But, if you have a mouse installed, at least in E5450 one can select "Disable Touchpad when USB Mouse is present", from the Mouse tab, from "Dell Pointing Devices", accessed via "Dell Touchpad", accessed via "Control Panel". Enabling the touchpad and selecting this option will get rid of future problems. Hopefully.
WOW! That is lazy.
I do this all the time, since i have a laptop that has a broken trackpad, and i need to use the Logitech Unifying Software to set up my mouse.
You will need a few keys:
Tab : moves you forward through selectable inputs (text fields, drop downs, buttons)
Shift-Tab :moves you backward
Alt : menus
arrow keys: duh
Enter or Space : Select, click button
Alt-Tab : to switch back to the app you need to be in
Windows key : then type the name of the app you want to use
Esc : Back or close window
I am able to do MOST things i need to do to get my mouse working again, you should have no problem getting into control panel or where ever and turning yours back on