After the much appreciated information and help I received here, I bought
a GeForce MX4000 video card and installed it. Booted up and it worked, with
everything looking the same as before the old video card went out. However,after restarting, all desktop icons were smaller but much longer looking, with those near the bottom (and the "start" button below them) going completely off the bottom of the monitor screen.

By moving the cursor arrow off the bottom of the screen in the general area of where the "start" button would now be, then clicking blindly around that area, I eventually hit the start button and the program display came up.

How do I fix this to make the new, taller display fit onto my monitor screen?

Any suggestions or help would be greatly appreciated.

Sounds like your graphics card is running a Refresh Rate that is not compatible with your monitor. Right-click a blank space on the desktop and go to Properties > Settings > Advanced > Monitor. You can normally adjust the Refresh Rate in there.

Also consider that you probably have controls on the front (or underneath the front panel) of your monitor to move the image around on the screen and re-size / shrink / stretch / twist / adjust it.