I would suggest calling and asking for the director of sales or general manager and explain your situation/dates you are looking for. I would advise against asking for a weekly rate, as far as I know Homewood Suites don't do this. It would just be some nightly rate x7 and asking for a weekly rate risks them thinking you may not be the type of client they want there and cause them to bump the rate up a bit or not offer a discount at all. Instead I would suggest using the phrase long term instead.

As far as what kind of discount that will really depend on the management company and their strategy. Some hotels will want to focus on long term guests (particularly if they run lower occupancy) where as if they have lots of week day business with a high rate they are unlikely to want to sell something longer as they will likely displace this. If you plug the dates into the website you will probably get a fair idea what the pricing will be as they typically put their strategy into the rates that are available. A phone call may obtain some kind of additional discount, but YMMV as to how large this is.