Ahhh.... Good topic, Van!
The "formula" for the perfect workout is an elusive one. To achieve that, we need to think about what you are trying to accomplish in the gym.
1) You want to burn calories. In the case of improving percent body fat, you want to burn more calories than you retain.
2) You want to promote muscle hypertrophy.
How you achieve that is an elusive thing, and that's a reason why many kids want to shortcut the whole thing and just take steroids. And even that comes to bite you in the arse at the end of the day.
The body is big on homeostasis. It wants to come to some kind of equilibrium based on the genetic instructions that came with it. And those instructions have been selected over other instructions over time based upon what life was like thousands of years ago. And life sure has changed a lot in just the past 100 years. Meanwhile, we're still carrying this outdated genetic code that was appropriate for another era. (It may become appropriate again if armageddon happens...)
The problem is the amount here. What people assume is that if X amount of work causes Y result, then 2* X work will cause 2*Y result, and 10*X work will cause 10*Y result. But it doesn't work that way. Your genetic code has been selected for based upon what really happens in the wild.
Let's take me as an example. I came from running track. I was a long distance runner. When I started martial arts, I was almost 6 foot (still taller than most who claim to be...

) and weighed all of 145 pounds. Why??? Let's look at the extreme. What does the average competitive marathoner look like?
Is this nature or nurture? Well... It's a little bit of both. Nature gave this guy slow twitch muscles, which give him great endurance. But the nurture of training made this guy's body pare the weight down to an absolute minimum necessary.
Does this guy look like he's going to win a bodybuilding contest? Hardly.
What happened to me when I quit cross country and started martial arts plus some isometrics? In one year, I gained 30 pounds of muscle mass. I wasn't doing anything special. I was just 18, stopped doing the catabolic (breaking down) long-distance training, and let nature (being an 18-year-old) do its thing.
Bodybuilders spend a lot of time trying to come up with the right formula for how to train to get big. There are almost as many formulas as there are noses in the world, so you have to take any written opinion in your average Men's Magazine with a grain of salt. However... There is a kernel of truth here.
To encourage anabolism (building muscle) and discourage catabolism, you want to stimulate testosterone plus growth hormone and minimize the production of cortisol. Cortisol comes out when you are physically and psychologically stressed. If it's just psychological stress and you treat your body like crap, you end up gaining fat weight - particularly around the midsection. If it's physical and you are on the borderline of that ratio of what you consume vs. what you burn, your body may decide to keep you thin, but get rid of as much muscle as it does fat. And once the muscle mass starts going down, your basal metabolic rate may go down with it. Your genetic code is concerned with famine, and doesn't want to put you in that state of depletion where you might starve and die if food becomes scarce.
There are rules of thumb for how to make this work right. There are rules of thumb for how to increase muscle mass and decrease body fat (rather than some "unfavorable" combination of the two). And in doing this, you have to remember that your body can be smarter than you, and will try to keep you from doing things that aren't good for it (according to your genetic code). Here are some rules of thumb on this.
1) Keep your blood glucose level constant. Don't drop it by starving yourself. Don't peak it by eating high glycemic index stuff (sugar, etc.).
2) Work out VERY intensely, but for just an hour. Do not go beyond an hour, or the end result will be less testosterone and more cortisol.
My personal recommendation would be to spend 5 to 10 minutes (max) doing your warmup on some machine, and then jump on the weights. You can achieve the equivalent of wind sprints just by doing less time but increase the resistance on the machine. Some machines have special formulas for doing interval training. Then work really, really intensely on weights using primarily multiple muscle group exercises. You want to shock it. Stretch in-between sets when the various mucle groups are warm. Then go home and enjoy your day.
This also means you have to find special ways (e.g. periodization) to change your workout so your body is constantly left guessing and "thinking" it needs to change to adapt.
You may want to do more aerobic work, but not on the same day as your weights. Maybe you go longer on a karate day. Do it before class to warm yourself up. Then your muscles will be physically warm for stretching. And your joints, tendons, and ligaments will be ready for some good hard work.
IMO, you are better off losing weight by concentrating on increasing muscle mass. It gets harder and harder to keep that with age as our bodies naturally decline in testosterone production. If we are vigilant, we can keep them looking pretty good much longer than people think. Even older bodies can produce testosterone and growth hormone in response to a good workout. And with the muscle mass higher, the basal metabolic rate is higher. As long as we eat right, you'll be burning more calories just by sitting and enjoying the buzz of your good work.
If you don't mind the look of a long distance runner, then by all means do more aerobic work. But there's a price to pay for too much of that. The lower the muscle mass, the less you can eat and not get fat. Some women never seem to get this...
Hope that helps.
- Bill