Donald Trump had a lot of help on the campaign trail. He was by no means central to his own success. He recruited capable individuals, including Steve Bannon, who knew how to play on the fears of the American public and secure the presidency for his boss.
In this respect, there is absolutely no reason why Donald Trump could not both win an election, and be mentally ill.
In the world of business and reality TV, mental illness can actually be an asset. His narcissism may have actually helped him become a strong reality TV personality. His outrageous lies would have made for good television, and his inability to stay on a single subject for too long would have made for a fast-paced show filled with twists and unexpected U-turns.
In business, mental illness is prominent among entrepreneurs. An estimated 72% of entrepreneurs suffer from mental health issues. Therefore, his business success, reality TV stardom and ascent to the presidency do not constitute evidence that he is not mentally ill.