You could make the sheet manually. It's not like you'll have to do it often. I do all my animations in Blender.

Sure these were done in blender as well as my avatar.

All of my sprites in this game were done in blender

But the animation in my game is an animated cartoon. So, every frame must be drawn in some software.

If that is the style you are going for exactly, then it will be that way. But, you can do things differently too. The picture in the above poster shows how creating 3d models for art can work for 2d games. You create the model(model, maybe UV/texture/animate too), then you take pictures(rendering) the animation frames as needed. Then those animations are put into a sprite sheet or loaded into the engine as a sequence, or however they are needed for your game engine.

If you are interested in doing pixel art, I'd use Cosmigo Pro Motion. For classic pixel art with a 256(or less) color palette, it can't be beat very easily. There are other programs, like anime studio, etc... that are more specialized for animation for cartoons. But you haven't been clear on exactly what you are doing, what art style, what type of game, so it is not easy to give a good recommendation. There is no single "best" tool, rather simply some are better than others at certain things, and regardless of that each person is likely better with one tool than another.

I liked the easy to figure out interface of Pencil. If I drew better I would certainly make more use of it.

Maybe take a look at Spine2D. I've not used it myself yet, but if I ever start a 2D game again, I'm most likely going to check it out. It's unfortunately not free, but definitely on the cheap end of software (~70$ for the base package). Doesn't seem like it would offer too much more than blender (which I agree with others, and definitely recommend), but may have a less steep learning curve and it has some fancy features to streamline other things like bounding boxes, skins, texture packing, etc.