The vehicle speed sensor shouldn't cause the brake light to be on. Check to make sure the brake fluid is full, and that the parking brake indicator switch is properly adjusted, and not damaged. Take a look through this link and let me know if it helps. Also, I am basing this on the brake light only and not the ABS light. If that is incorrect, let me know.
Next, the VSS can cause an issue with the speedometer. However, it could also be a wiring issue, connection issue, PCM issue, or instrument cluster issue.
If you have the new part, I would install it to see if it helps. I attached the directions below for replacing it.
If that doesn't take care of the problem, scan the computer to see if there are any diagnostic trouble codes stored. Here is a quick video showing how it's done:
Let me know if this helps or if you have other questions.
See pics below.
Images (Click to enlarge)
Tuesday, November 30th, 2021 AT 7:21 PM