Quote from llg
It would definitely be unethical to ask your current employer for money now. You agreed to do a certain job for specified compensation. Based on your agreement to work for them, they paid you while they provided you with valuable education. For you to "welch on the deal" now by asking for more money (or by leaving) would make you look very bad -- and give you a bad reputation that might come back to haunt you in the future.
I would only consider giving myself a bad reputation under extreme circumstances -- for example, if you found out that your current employer had lied to you about the job, or had done something that was very wrong, or turned out to be a horrible place to work, etc. Unless your current employer has done something wrong, it is very wrong for you to take their money (training) and run.
Why would it be "unethical" to ask for more money? They're big boys and girls, and they can say no, if they are so inclined. But showing them a firm offer from the other place just might induce them to pay more. It cannot hurt, and might get this nurse better money from the deal.
The current hospital hasn't provided "education" (unless there's something I'm missing); they've done orientation.
Having said that, nurses should know exactly what they're getting into with a sign-on bonus. A hospital doesn't do that unless they have trouble getting nurses without it, and one has to ask why they can't: is it bad working conditions, a bad physical facility, a very bad neighborhood, or what?
Also, other posters have mentioned burning bridges. This is correct. Burning bridges is not necessarily a bad thing (some bridges need burning ... :spin
but it's unlikely you can go back once burned.
Another option would be to take the second offer, resign from the first, work out a notice, and be gone. Hospital # 1 wouldn't be excited, but would probably not be too unhappy.