I would not tell her anything. If you ex-colleague is willing and able to keep doing "most" of her previous work, I would let her do it. Unless you have not enough work to do, or she doesn't do the work well enough, I would not make a problem out of this.
Also, there may not be enough work for her to do in her new job. Just stand by and wait until she gets a better grip on her new position and then slowly take over her previous work.
In the meantime, enjoy the simple life.

I totally agree. One thing I would add is that you may consider informing your supervisor of the challenge you are facing and your plan to rectify the problem before you approach your co-worker. By taking this path, you will not only have your supervisor's support but you will show him/her your ability to problem solve proactively.

It is not up to you to "tactfully" tell her anything. You should approach your direct supervisor and let him/her know what is going on. They can then approach her manager and come up with the best way to help her transition into her new position.

I was in a simular situation once where the other Admin was no longer doing her old job duties really, but we did have to change the password on the database that we were using because she couldn't stay out of it, we know this because the computer noted the names of those that last made changes. But she refused to give me all the files and things that went along with the job. When she trained me on several software programs that were used in that department, she purposely left out steps and things so that I would make mistakes. (The main reason they reassigned her was because she was making alot of mistakes and didn't mix well with the manager of that department.) Though thank goodness I am very smart so I figured things out on my own very quickly. I went to my supervisor about getting the files from the years past that could have been a help to me, who said we just have to be patient, try to be understanding with her, she will come around with them. But his wishy/washy idea did nothing but let her have her way. Only when they ended up getting rid of her like 4 months later, was I so surprised to actually see the huge box of things that she hid from me that pertained to my job, but by that time, I ended up having my own contacts, and creating my own forms and things that I no longer needed any of those items. lol

I was in a similar situation as well. In my previous job I replaced a woman who retired & was not really very technology savy. Neither was the other admin in the office, but the other admin "knew" how to do the things the other woman did because she was the back-up for this woman for many years. The other admin was very inefficient in general. As with Kimberly's situation, the other admin trained me, but left things out. Then when I did things the way she trained me she was run over & tell me I did it wrong. After a couple times of this happening I spoke to our supervisor about the situation because I didn't want it reflecting badly on me. She also had a habit of trying to blame things on anyone else & not take responsibility for her mistakes. A responsibility was taken away from her and given to me, she seemed to think I was not capable of handling it. What she didn't know was my boss knew very well I could handle it. My boss had to tell her it was no longer her responsibility & she was not to handle it any longer. I definitely think it is something that should be dealt with through the supervisors. You want them to be aware of the situation so if mistakes are made they don't reflect badly on you.