Load up the parts cannon!
There are many possibilities. Troubleshooting is needed. It could be something as simple as corrosion on one or both battery cables at the battery terminals. The "whine" you are hearing we can't hear... it could just be the blower motor running for all I know remotely.
Since you have a voltmeter, you can modify NW-Bounds simple test in post #3 - Put the voltmeter probes on the positive and negative
POSTS of the battery.
NOT on the battery cable clamps! Press hard to make good contact. Read and note the voltage. Then while holding probes on the battery posts, have someone turn the key to Start. What happens? Does the battery voltage measured at the posts go way down while in Start, or does it stay approximately the same? If approximately the same, either there is corrosion at battery terminal(s), or bad ground, or starter not being engaged due to some other wiring/switch problem. Or the starter motor's commutator is wearing out, the old tap the starter with a hammer to jiggle it often works for that, but is very temporary!