Sorry to hear that Marquis is having problems. To start with the bottom line, he needs to be seen by a veterinarian that treats reptiles. The veterinarian will need to do a physical exam, check him for parasites and do blood tests to figure out why your monitor is not eating. Of course, even without eating, the lizard will still pass wastes. The urine is passed as a solid, white crystal and that will be seen even if there is no brown stool to pass with it.
Lizards needs a temperature gradient to live in.
In other words there should be 'hot' and 'cold' areas of the environment that they have access to at all times. This allows them to choose what area they need to keep their body temperature in the right range. The diet that you are feeding is not adequate for a monitor. I know that he is not eating anything now, but chicken, beef hearts and liver are not good to have as part of their diets. The rats are good.