Buckworks, congratulations on the achievement! I did keto for years and it worked well for me, but it did cause some hormonal issues which have been outlined here over the years. In short it: there is an inverse relationship with SHBG and insulin production in the liver. My insulin was so low for so many years that my SHBG was pushed out of range and my testosterone was being gobbled up at higher numbers than it should have been. Fast forward, I am injecting myself twice a week for the rest of my life. I find the benefits very much worth it though. There are reasons why I think this is less of a concern for women than for men, so I don't think you need to be very concern about a rise in SHBG messing with your hormonal balance.
I really encourage you to do resistance training when on Keto, preserving muscle mass is very important and also very advantageous to losing fat on the diet. You don't need a gym, you could get tremendous benefit from just some simple equipment and working out at home. I don't know your exact age, but I think you are probably somewhere in your early 60s? When dieting on the plus side of 50 we need to really think about sarcopenia. Avoiding it has shown to extend life and extent the quality of life.
Here is what I advise for gear:
dumbbell set (I don't know how strong you are, but you may want to start as light as 2.5lbs or 1kilo)
a stable four leg chair height stool
a floor mat big enough to lie down on and with cushioned enough to be comfortable to do floor work on
a step, as in stairs or the kind they sell for workouts
That's enough equipment to work your entire body.
I don't know if you have any nagging issues with rand of motion or curtain types of movements but I am going to outline some basics that when done together will work your entire body. You don't have to do this in any particular order but I usually start with the larger movements first.
Pushups (shoulders, tricep, chest, core)
If you are strong enough do regular push-up. If those are too hard do bent knee push-ups (what they use to call girl push-ups when I was a child). If those are too hard go to a wall and do a push-up against that. A lot of people make a mistake when it comes to push-ups and have their elbows out too far, they should be close to your body, It is harder that way, but better for your shoulder joints. Your hands should be slightly turned outward too so that your thumbs are up.
Goblet squats or Lunges (quads, gluts, some core, balance and overall body strength)
Goblet squats are easier than lunges. Hold a dumbbell at your chest and squat down. If that is too hard you can do goblet squats on and off a stool, You can also do it with no weight at all and use a stool with no weights too.
Lunges are great, you can do them with a dumbbell in each hand or without any weight at all. If you don't know how to do them
properly
here is a video of the basic movementCave raises
If you have stairs with a sturdy railing use that for support. If you don't have stairs get your step and put it next to a wall and use that for support. Step on the step and stretch your calves then lift up on the balls of your feet, and repeat. As you get stronger you can do this with weight in one hand and/or do one leg at a time.
Straight leg dead lifts (hamstrings, spinal erectors, lower back)
Hold the dumbbells in each hand and bend down at the waist while keeping your legs straight. You want to go easy on this at first. You can do it with no weight for this movement too. Always do this movement slow and controlled.
Dumbbell shoulder press (Shoulder muscles, triceps, trapezius)
While standing have the weights in your hands at about your shoulder level, slowly raise them above your head and bring them back down.
Floor Ab work (core)
You could do planks, crunches or leg lifts. Crunches are probably the easiest to start with. Do them nice and controlled. Make sure to have that mat underneath you.
One arm Bent over row (Upper back)
You will have to be bent over with one arm on that secure stool (you could use a chair too) and have one arm at your side hanging down. You want to keep that elbow close to your body. Bring the weight up to where your ribs are and then back down. Concentrate on where your elbow is moving not on where the weight is. Try to not round your back but keep it straight and parallel to the floor. A lot of peolpe tend to do this one wrong and use their biceps to curl the weight up in the movement. Your bicep should not be engaged at all and should be relaxed. Obviously do a set on both sides.
Dumbbell curls (bicep, wrist strength, grip)
Move slowly, don't swing your arms around so that your elbows are moving, They should stay at your side. Bring the weight up it is approaching your shoulder and then bring it down. You could alternate between the arms or do both arms at the same time.
Bent over side laterals (back of the shoulder mostly)
I am throwing this in there because people tend to get imbalanced in their shoulder development and that could lead to injuries or joint problems. It is a simple movement. You cold do it bent over with both arms at the same time, but I don't recommend that at first. Use the stool instead and do each side separately. The support of the stool will help with form and keep you level. You don't want your back twisting when you do this exercise. Your back should be flat like when doing the bent over row. In this movement you are pulling the weight outwards away from your body so that it aligns with your shoulder. Your elbow should have a very sight bend in it the entire time.
Here is an example of someone doing it standing. I don't think you need to do any tricep specific exercises because they are being worked in the shoulder press and the push-ups. You can add them though if you want to.
If you do all this you basically are working your entire body. There are many exerciser you could swap in or out to target muscles in different ways. Everything I outlined though are relatively easy on joints and can be done for years without issues if you are sensible about your movements and the weights you use.
I recommend doing just one set of each exercise per workout and do the workout at least three times a week. You could get the whole thing done in about 30 minutes. I know this is long, but you wanted advice and I seem to have a lot to say on the topic. If anything is confusing, please ask questions.