Need to change that sentence into, "Wife is overweight and wants to lose fat by finding a sustainable and enjoyable way through diet and exercise." The whole "as fast as possible" is setting yourself up for failure and these types of things need to be treated as literal lifestyle changes and adopting a different way of living. Her habits or choices got her where she is, these need to be changed slowly over time.
Arthritis-- exercise can help this, but again needs to be done slowly over time. Can she walk? Start by walking. Most of us are self conscious and she needs your coaching to help ease her into being active around people. Not all gyms are created equal by the way... like I would never willingly train in a Gold's Gym, I don't like the atmosphere. Have her do push-ups against the wall or elevated, make exercises easier with the goal of eventually doing them fully one day in the future.
Have her learn how to eat sensibly. If she is overweight, she doesn't need to do any fancy diets. She just needs to learn how to eat healthy and to make good healthy food choices. That is some research on you twos part. To get you started, eat whole foods and nothing else. Learn how to cook well at home. She needs to eat well and needs the proper foods to help her lose weight and be healthy. The meds aren't doing what you think they are, it is her habits and lifestyle choices.
It is really simple man. Be active every day and eat a healthy diet. Done. No fancy workout programs or diets. Doesn't matter what you do. Go to the park and walk, ride bicycles, swim, go hiking, go play miniature golf, clean the house, do yard work-- be active and eat a healthy diet. All of this specific fitness stuff when it gets sort of whacky and complicated is for people like me who are trying to squat three times their body weight and trying to build unusual amounts of muscle compared to the average person. If you want to be healthy and just look like a healthy human being, just be active every day and eat a healthy diet 95% of the time. To achieve this, you both have to find things that you enjoy doing that enable you to physically work your bodies and fuel your bodies with quality whole foods. Over time, her body will want and be able to do more and more and thus a stronger body she can create-- but she has to find enjoyment in it...