I think you should polish up your résumé and start looking for a new job.
Yes, in an ideal world, you should be training the new person while you are being trained for a higher-level job. That clearly isn't happening here.
There is some reason the boss wants this person in your job. Perhaps these are good reasons, perhaps not. Often we are the worst evaluators of our own performance. Everyone thinks they are above average and doing a fantastic job. Bosses hate to give negative feedback, so this idea persists until one day—wham, you're demoted. It's the wimpy, ineffective way of managing. Unfortunately, it's also common.
While you are looking for your new job, take some time to think of what you'd like to be doing if you stayed at your current company. Then, make an appointment with your boss and apologize for writing that letter complaining. Why? Because by sending it to the company president and HR, you tried to jump over his head. This will make him angry. (No one likes to be tattled on, so before you take that route, make sure you are willing to deal with the consequences. Sometimes, it has to be done. I'm not sure this was one of those times.) Then, ask him about your career planning: "I'm very interested in moving into X role. What things do I need to do, and what skills do I need to obtain, in order to move in that direction?"
Note that you don't say: "I'm much better than the woman you gave my responsibilities to!" Leave her out of it and don't mention that episode again. Ask what you need to do. You may find out the reason for your demotion is you. Or, you may confirm that your boss is a jerk. But, take the high road. And if there isn't a clear career path, then keep looking for a new job.
Jobs are out there. And to be honest, a good administrative assistant is a highly sought-after commodity. So, market yourself and find a great new job. But make sure you have the conversation with your boss. You don't want to accidentally make the same mistakes again.
Suzanne Lucas has nine years of Human Resources experience, most of which has been in a Fortune 500-company setting. She holds a Professional in Human Resources Certificate from the Society for Human Resource Management. She blogs at Evil HR Lady.