Heathrow Airport to Victoria station hotels

When you arrive at Heathrow, you can take the Underground "Tube" to Hammersmith, where you can change to the District Line that will take you the rest of the way to Victoria: https://tfl.gov.uk is the website with all sorts of info on London transit.

I really like taking buses to get around in London. Try them out if you get a chance. They are a little slower than the Tube, but have a great view from the upper deck.

I am not aware of a B&B or Hotel near Victoria in your budget. I like the Lime Tree Hotel. It's more expensive, but if you can swing the extra cost, I highly recommend it.

When you leave to Gatwick, you take a train from Victoria Station, not the Tube.

Answers to several of your inquiries...

Cheapest, easiest, to Victoria station is the underground which you can catch from any terminal, but you likely will be coming in to Terminal 4. As the previous poster stated, take the Picadilly line which leaves from the Airport to Hammersmith station, then get off and walk across the platform to the District line that will take you to Victoria. You can also change at other stations, but that usually involves steps and changing platforms, Hammersmith is easiest. There are also buses from Heathrow to Victoria Bus Station, those would be direct.

For hotels, you just need to look around. For your budget, any of them would be about the same. You might save a bit by staying out a bit further out. I have stayed several times at the Adria hotel in Hammersmith, economical, quiet. I have also stayed in Earls Court area, also right on the underground line, which has a variety of options.

Military Discount? Probably not, nor AARP, or Senior, or other US based discounts; just not their thing.

From the Victoria area, yes there is a direct train to Gatwick, also express bus options,

We stayed at the Cherry Court hotel in May. We loved the location. It is very close to Victoria Station - within a block or two. The rooms are very small but clean and nice. They give you a basket of fruit and granola bars every morning. I think we paid 75 pounds per night.

Try the Union Jack Club near Waterloo
http://www.ujclub.co.uk/

The Union Jack Club is for serving non commissioned members of Her Majesty's Forces and active serving members of other military forces. If you are non-UK retired, or active UK above the rank of a non-com you can't be made welcome.

I was trying to stay in the Victoria Station area because it is the easiest way and cheapest way to get to Getwick airport when we leave if there is any suggestion on other area's that would be great but we really have only 3 days in London because we get in so late the first day. Should we stay near Heathrow the first night and then go into the city the next day to stay the time I know there is a hotel's right near the airport if we can't get the tube?

I would advise the OP to check, though. There have been posts on the London Forum of Trip Advisor with current and retired military families from various countries who have booked there, most recently June of this year.

They have a Facebook page and Twitter account, so you can ask via those channels also.

Some hotel chains could possibly offer military discounts through their USA sites I guess. The UK schemes tend to be restricted to British Armed Forces with those from NATO stationed in the UK sometimes in addition.

Although members of other forces cannot become members of the Union Jack Club they do allow them to stay in the club as a Temporary Honorary Member albeit at a higher room rate than members.

We just stayed 4 nights at the Premier Inn near the Victoria station...maybe a 3-5 minute walk...several nearby places to grab breakfast...Premier Inn has nice rooms, air conditioning, and importantly, lifts (elevators)...within your price range, but Saturday nights are more expensive...breakfast buffet is available in the hotel

While arriving from Heathrow airport, taxis or buses to reach Victoria station area. There are a number of hotels in the area. You can search it in search engine. To travel from Victoria station to Gatwick airport, it is recommended to use trains for the journey. Happy journey!