I've never been on the guest side of this, but I used to be a Revenue Manager. Here are a few thoughts based on my experience.

Do not call Hilton Central Reservations/Corporate. Each individual hotel is responsible for their own profitability. Literally no one at Hilton has the authority to negotiate rates.

Call the hotel and ask to speak to someone about a rate for a long-term stay. Who you end up speaking with will vary by hotel. You may end up speaking with the GM or maybe someone in Sales. You may also get forwarded to the company that owns the hotel. (Some companies own multiple hotels within a metro area. Typically, there will be one Revenue Manager who oversees multiple hotels.)

This is a really difficult question to answer. Most hotels will plug the dates into their revenue management software to come up with a rate. The software will spit out a break-even number based on projections. Without access to data, it's hard to know if you're getting a good deal or not.

Although a six week stay is significant to you, keep in mind that 42 nights might not be that big of a deal to that hotel. As scubaccr notes, the only benefit the hotel gets from your stay is on days when they aren't projected to sell out.

It helps to put your stay in perspective against the total inventory at the hotel. For example, a 200-room hotel has 73,000 room nights to fill each (non-leap) year. A six-week stay in one room is less than 0.06% of their total inventory. There are very few businesses that would provide a substantial discount to a customer over such a tiny percentage.

I'm not saying you might not get a great discount. But, I wouldn't be surprised to hear that the hotel declined to negotiate.

I agree with scubaccr here. The hotel will almost certainly want a guarantee that you will stay the entire time.

All points/nights/stay credits are negotiable. If you want them, ask for them.