Since there are different levels of mental illness, there are different levels of challenges as well. However, there are some challenges that would be universally an issue, no matter what type or level of mental illness is present.
One fairly obvious challenge is that of patient advocacy. In many cases, the patient can be their own advocate, but in the case of mentally ill patients, this is often impossible. A person who is mentally ill is often incapable of standing up for their rights (or even knowing what their rights are), challenging a faulty diagnosis or treatment plan, or reporting physical, emotional, or medical abuse. Imagine a patient who is schizophrenic with moments of lucidity trying to report that someone has abused them. In many cases the abuse is imagined or grossly exaggerated, but what if it's true?
An immediate challenge that would be faced by a mentally ill patient in a hospital would be in getting their needs met. A patient may be unable to communicate that they are cold, hungry, lonely, or in pain. This presents a challenge to both the patient, and the medical practitioners charged with their care.
Hospitals can be frightening places for every human being, even when you are capable of receiving explanations of new or even painful procedures like CAT scans, x-rays, shots, etc. Imagine being a mentally ill patient and having these procedures done to you with no idea of what was happening, why it is happening, or how long it will last. Fear is a huge challenge that many mentally ill patients are forced to deal with while in the hospital.