Caroline Miles gives guidance on how to discipline an employee for misconduct.
The purpose of the investigation and the disciplinary hearing is to decide what, if any, disciplinary action is appropriate given the circumstances.
If following the investigation there is no justifiable reason they got the keys cut and you are suspicious of the reasons for doing so, then it could potentially be a gross misconduct issue (a serious breach of trust and confidence perhaps?) and subsequently you could move to dismissal, but you need to give the employee the chance to state their case and explain the circumstances as there may be a simple, genuine explanation (alternatively, they may not realise what was wrong with what they have done and just need a warning).
Regardless of length of service, the statutory minimum dismissal procedures should be used if dismissal is being contemplated (i.e. invite to meeting, hold meeting, write to appeal etc) to avoid any future come back even if they are not within the qualifying period to be able to claim unfair dismissal.
If, as you say, there are performance issues – this should be dealt with as a separate issue and not disguised as something else (i.e. don’t use this as an opportunity to get her out if the real reason is performance). You can performance manage her and dismiss if necessary through the disciplinary procedure.