I interpreted your character's lines to mean she wouldn't leave any money to her doctor in her will, as that might provide an inducement to overdose her! Inducement is therefore the correct word, as inducement can have the connotation of tempting someone to do wrong. To provide someone with an incentive is to get him or her to do something that is of benefit to you and/or her or him.
The Environmental Protection Agency, for example, provides incentives of various types to polluters by allowing them to address and fix their violations before the EPA cites them with administrative fines, or civil or criminal penalties. The EPA's congressionally-delegated authority to offer incentives to polluters benefits both the polluter, on the one hand, and the environment and the citizenry, on the other hand. Persuasion can also play a part in incentive-based compliance. In psychological terms, an incentive is called positive reinforcement.
An inducement, on the other hand, can also be used persuasively, and its purpose is also to get someone to think and/or act in a certain way. Take the financial consultant who induces you to invest money through his company. If the company has a long tradition of getting its investors a good return on their investments, then you are more likely to succumb to that positive inducement (or incentive). If the financial consultant induces you to invest in a Ponzi scheme of which you are not aware, then you have succumbed to negative inducement. The scoundrel will certainly provide you with various incentives to invest with him by dropping the names of satisfied customers who have made millions with his help, or by sending you an incentive check a few weeks after you have given him your money.
Interestingly, according to Google's Ngram Viewer, in 1800 incentive was somewhat more common than incentive. Starting around 1900, however, incentive took off steadily in an upward direction, while inducement lost ground steadily.
You could say with some accuracy that inducement is perhaps a more-versatile word with both positive and negative connotations, although I'm almost positive there will be folks who disagree with me.
A final note. While a person can induce a person to do something, she can't incentive you to do something. Additionally, you can say, "I was induced to invest in his Ponzi scheme," but you can't say "I was incentived to invest in his Ponzi scheme."