Safety and children go together I feel to a certain extent. However, I also feel that children need a chance to be children and explore their world (within reason). It can be tricky to find the right balance, but any risks I take, are calculated risks based on my experience with that child.
In your particular case, the child at most risk is your 9 month old baby who is now mobile. Your baby will want to explore their environment but restrictions will need to be put in place to avoid accidents.
"she could sit inside on the sofa and watch her" - you are not paying your nanny £10 per hour, plus gym membership (thinking back to your other threads) for her to be sitting on the sofa watching your baby play outdoors.
Yes, you should make the deck safe... until it is safe, then your nanny shouldn't be letting the children play there - especially leaving a baby unattended!
I would say that you should trust instincts. You need to have a childcarer whom you trust. While they may make mistakes on occasion (after all, even a supernanny is human), you don't want to be constantly worrying about your children's safety and well-being.
It will certainly be easier to end your employer employee relationship sooner, rather than later - as for the first few months it could have been a probationary period.
With regard to a contract, you do have a contract with your employee, even if it isn't written. Most employees are entitled to 1 Week notice after 1 month's service. The DTI say that "All employees taken on for one month or more are entitled by law to be given, within two months of the date the employment starts, a written statement setting out the main particulars. This statement will not necessarily cover every aspect of the contract, but will constitute important evidence of the principal terms and conditions." More information about Contracts and Written Statement from DTI.
As you say you do not have a written contract, then I feel it will be hard to dismiss your employee on grounds of Gross Misconduct (as you haven't set out what grounds those are). Therefore may be better to terminate with 1 week notice period (unless you had previously agreed any longer period of notice, verbally or via e-mail).
Should you decide to keep this nanny on, you must produce a contract/written statement, and I would strongly suggest that you include a lot of information in the Gross Misconduct section, especially about safety - so that your nanny is clear as to how important you feel safety is, and what actions would result in instant dismissal.