As a former nanny and with a couple close friends who have been nannies for years there are some really important things to consider when hiring one. Soo
first for the firing of your nanny. I would be really up front with her and tell her you are letting her go. However, I think it is really unfair to fire her without any notice at all. Even a week would be alright and honestly if I had been fired and they asked if I could or would work for another week or so I would have just politely declined. If you are concerned for your children's well being because she would handle it badly that is another matter entirely!
As far as questions to ask like the PP said they vary greatly by your situation. It is important to set very clear ideas about what the job entails. For example, do you expect cooking to be done? How often? What kind of food and are there any food restrictions that you need to relay to potential nanny? Do you expect housework to be done and if so what kind and how often? Do you expect your nanny to care for any pets at the same time she is watching your children? What schedule do you need your nanny on, are there times when she will travel with you? Will you expect her to work without notice if you need her to? Are there any restrictions on specific activities, ie. no television for the kids?
All of these are fair things to discuss with your possible nanny because it is really really frustrating from the nanny perspective to be hired and have a good idea of what you think you will be doing only to be lectured when you don't do the laundry or something when you didn't know it was required!
For example, I worked for a family for a summer where I was told I would need to do light cooking and one load of laundry a day as well as take the 2 dogs out for a walk once a day. Well that turned into me being questioned as to why ALL the laundry wasn't done every day and why I hadn't cleaned the dogs poop up in the back yard and why had I let their daughter have a particular snack that was in their fridge anyway. I quit that job pretty quickly...
See what I mean about needing to be on the same page though? It is SOOO VITAL that you are because it really prevents headaches for both sides in the future.
As far as your possible nanny meeting your children that is also very important to me, because frankly and please don't take this as a slight against your children I have interviewed with families and met their children and from meeting the kids knew it wasn't going to work out! Also you want your kids to be comfortable with a possible nanny and it is better to find that out before you hire someone.
Finally, sorry for the rambling, please do a CORI on whoever you choose to hire, for your families sake. This should be at your expense and it is really quite easy! Also as far as pay goes I know you said you had a general idea but IMO as well as every other nanny I have ever met in my life (quite a few we travel in circles

) $15 an hour for one child is the bare minimum I would accept as payment. More for more children. This may seem like a lot but you have to ask yourself, you are trusting this person with your child it is not an easy job at all. It is stressful at times and can be a lot of work (think about how hard it is to raise a child, it isn't the same but you are still asking someone to care for that child and act as a guardian when you are not there). Honestly I would question someone who was willing to work for less than $10 an hour!
I hope I didn't ramble too much and I hope some of that helped! Good luck and be patient with your search it can be really hard to find a good match, I know coming from the other perspective!
sorry I just read your update and am glad that you were able to give your Nanny that you are letting go such good treatment! It means a lot it really does.