There are plenty of these chips. TI make some. However, they are designed for professional circuits. Resistors are easier.

 

Is there an off-the-shelf chip that drives LEDs by current-regulation using a DC-to-DC switched-mode converter, using only a low-resistance sense resistor (internal or external) to sense current?

As has been mentioned, major overkill. There is a myth trying to form that resistors are not adequate, this is false. LEDs have used resistors as long as there have been LEDs

Here are some possible arrangements with 9V and 6V (4 AA or AAA batteries)...

9V batteries discharge to 7V pretty fast, so a single LED/resistor may be better. AAA or AA batteries are deeper, and will last longer. The schematic using 6 LEDs will use the exact same battery power and the first schematic.

 

Some LED Driver ICs.

You can learn what we all learned from LEDs about current and voltage.
You can also look at a few IC solutions. Yes they are overkill but if you want to make some really fancy flashlights then you might find an IC that gives you a long battery life and uniform brightness from the fully charged to the nearly dead battery, and that is a trick that you will have a hard time doing with just series resistors.

LT3491 from that link seems to be a good choice for your flashlight.

 

OK, so first off I just wanna say thank you to everyone. I figured out what happened the other day when I tried it again (on purpose this time, concentrating on the LEDs to see what would happen.) I was giving them too much current, as they did stay lit for 5-10 seconds, but they had a green/blue glow to them, then one by one, they all dies out in about a second or two. Just too much current. I am still learning and still experimenting, but I am actually glad this did happen. I was just thinking about voltage and now I have learned that voltage is not the only thing you can look at, there is more too it. So thanks again to anyone and everyone who helped!

 

The odd coloring you are seeing is caused from overdriving the LED.

Its another thing to add to your list of things learned. If you notice this on later, you will know that the overdriven LED has a current limiting problem.