I'm going to answer the first part of my own question.
After upgrading my iPod Touch to iOS 5, Location Services started working again. Not only did it start working, it's showing the location as a point in the street in front of the house.
1) The location I manually submitted to Skyhook in 2009 was in the street farther south, by the mailbox, so the old data point is not being used. Not suprised since I have no reason to believe that Apple went back to Skyhook. A visit to Skyhook's website verified that this point has not changed.
2) The location that Apple started using last year was a point on the main highway 1/2 mile east of here.
So somehow Apple has independently obtained new data about my wifi location. It's possible that during a recent party, somebody with an iPhone had connected to my Wifi and that data was sent back to Apple. It's also possible that a Wifi data collection vehicle drove down this road. It's very rural with only three homes on a one mile stretch, so although possible it's difficult to believe.
It was my previous understanding that Apple started requiring that devices be within range of several Wifi signals to improve Location Services accuracy. Being in a rural location, it's nearly impossible for my iPod to pick up any Wifi signal other than my own. With Skyhook, this was never an issue.
However, iOS 5 seems to heavily depend on Location Services so maybe they've changed something critical to how it functions in order to improve accuracy without requiring multiple Wifi signals.
I'm going to accept my own answer. If anyone has a better answer to my original question, I'll consider accepting that instead.
You are most likely out of luck here. Essentially, you can not install any drivers on the iPad, and the handbook states on page 69 that...
Drivers are needed to use either the "USB tethered mode" or "GPS over
Wi-Fi" features.
This question mentions a jailbreak for iOS to use a bluetooth connection to pass GPS data into an non-3G iPad, but your hotspot does USB and WiFi only — and I personally would not advise to jailbreak for certain reasons.
The most promising approach to your question is this: If you do own/have access to an iPhone, you could try to tether the two via Personal Hotspot. The iPad might get the location from the phone — as described in this article. Let me know how this works.
Best Answer
Granted you have an iPad with GPS chip, the location services do work, but the iPad may take a bit longer to acquire the correct position. Apple explains this in a nice support article.