Hmm, tricky one.
Based on your description of nanny, length of service, type of relationship with family, etc., I'd say that she is very comfortable and has gradually taken liberties which slowly have become more serious and/or questionable.
She has been dependable and reliable and has essentially been an extended part of the family. The lines have gradually blurred, and now she has overstepped the mark in a big way.
I doubt she should be classified as an untrustworthy thief - as evidenced by the fact that she came to you straight away and told you about the use of the credit card. I suspect if she had asked you for a short term loan, you would have agreed. But she was very wrong to simply take without asking.
It sounds as if there is a good bit of money sloshing around (full time nanny, holidays, horses, etc) so I suspect pennies are not pinched or accounted for in any great detail in your household. She probably is using your horse feed for her animals - so while not overtly stealing, is not being scrupulous about what you are paying for vs what she is paying for (or should be). And until now, it hasn't been too much of an issue. She has gradually got too comfortable, taken too many liberties (too many times), and now has gone much much too far.
Think you don't need to let her go, but do need to sit down and have a very serious conversation. Give her a warning. Tell her she's seriously shaken your trust in her, and let her know exactly what is/isn't acceptable (as obvious as it may seem).
You can decide how you want to handle money with her in future - kitty money (with receipts - even if you don't check them she will be accountable), a pre-paid credit card (again, with receipts), or if you wish to continue letting her use your card.
It is one thing to be generous because you can be (and want to be), but quite another to be taken advantage of. No one likes that, and it is unacceptable. What she has done would be a sackable offense if she was newly employed and didn't have a long (and positive) track record.