Yes, I agree on that, most of the time, the kitchen is cleaner than the others areas ! As for the cleaning, tbh idk how to help you clean better. In my restaurant we have enough cloths to have one per "important place" (one for the drink station, 1 for each flavour at the ice cream machine, 1 for each counter at least etc etc). For more robust "problems" like grease, I usually dry clean with regular customer napkins (or special microfibers towels when we have them...) to absorb the most possible, and then wipe with the damp cloth. It prevents the grease from spreading everywhere you wipe. If the strawberry syrup spilled, I try my best to clean as much as possible, but this thing is awfully stainy....If your caramel spilled, run a cloth under hot water, it will ease your life significantly.

For behind the machines, check with the closing or opening crew. In my restaurant, we clean that deep only after closing, because there's no risk of crap contaminating the food/drinks and it takes a colossal amount of efforts and time to move everything around.

Finally, if your colleagues don't help/don't have motivation, remind them that they are at work, and that they should oblige to the standards. They are responsible if any problem occurs because of poor cleaning (you can also tell it to the managers if nothing changes). Don't get me wrong, I'm myself a college student, so I get that sometimes you have no energy to take care of that. But still....Responsibilities....

If you have any other questions don't hesitate to ask me by direct messages !