As another commenter said the model number is inside the door, you probably will need to remove screws from the back panel but getting it out of its hole should be relatively easy, most dryers weigh 100lbs max and are pretty easy to manhandle. If you search the model number you can likely find a replacement part online. What does the knob do? If it's the timer the replacement units usually run around $100 USD.
Did you look for a (usually silver) sticker on the inside of the dryer door? That's traditionally where the model number is applied.
I think you can manage. It will likely involve: moving and unplugging the dryer, removing some screws (probably with a 1/4" nut driver), undoing some snap fits (You'll need a thin, firm tool like a paint scraper or ruler), and plugging the new dial in. Nothing too complicated, and the whole process should take around 20 minutes.
If all of that sounds doable to you, I'd suggest starting by searching youtube for "[brand name] dryer control panel" to get a better idea of how to pop that off.
If it's an older unit, disassembling is very doable and you can probably just get a little hemicircular tube of metal of the right dimensions and then run with that.
Make sure to unplug it first, of course.
If it's a newer model shit might get complicated because there may be a PCB nearby and it is more likely to have berm assembled with uncmon screw heads (torx or something) or snap-on plastics.
I'm surprised there's no brand! I'd expect to find a sticker or something in the bottom or by the power cord.
Others have good advice here; my only add (which you may have already surmised) is that youtube is a wealth of advice for this sort of thing
Just tinkering around with various searches may let you know what is feasible and what isn't
So.. I've tinkered with my dryer a few times. Its a model that is pre 90's and all it took to get most of the panels off the sides, top, and the where the controls are was a phillips head screw driver, a flat head screw driver, and a 1/4'' socket ( I think) and its socket wrench.
Does the "nubbin" feel solidly attached but roughed up? You can probably gorilla/super glue a hollow shaft of wood or hard plastic (maybe a cheap Bic pen?) to the nubbin and the roughed up section will give the glue some crannies to adhere to.
Possibly one or two very small hose clamps if glue isn't going to work or the hollow shafted item is too large. Try to cut one side of the tube, very carefully, so that it will expand over the nubbin (which you might need to smooth out) and the hose clamps will hold everything together. Then just find something to attach to the hollow tube as a knob.
Are there local appliance parts/repair places? They might have a serviceable timer and knob that will work with the dryer if you need to go that far (and it will likely be way cheaper.) Might need to know the make/model of the dryer first.
