A You should not confuse these two separate issues; if you wish to take action, you should decide which route you wish to follow and not try todo both at the same time.
Taking illness first, it is not unreasonable for an employer in a situation such as yours to ask for a medical report and to ask what the prognosis is for a return to fitness. While the employee may decline to provide this, the refusal might be taken into account by the employer in determining the course to be followed.
If the condition is chronic and a return to full fitness is unlikely, it might be prudent to put the situation to a representative of the union to which this teacher belongs and ask the union to investigate whether this might be a case where retirement on the grounds of ill-health would be in the best interests of their member. An approach through a union is much less threatening to an employee, especially a sick one, than a direct one from the employer.
If that initiative succeeds, and assuming the medical evidence of incapacity is sound, your problem is over. If it does not, there are two possible outcomes. If the teacher is unwilling to accept this solution and the absence continues beyond the point where the period on half-pay has expired, your governors might consider issuing the contractual notice to terminate the contract on the grounds of frustration. This is simply saying that the teacher had a contract to teach at the school and has not fulfilled it for so long that the employer is justified in declaring it to be at an end.
Assuming that she has at least two years of continuous employment, the teacher has the right to claim unfair dismissal before an industrial tribunal, but if you have evidence on record of sympathetic support and an attempt to secure retirement on health grounds, it would be unlikely to succeed.
If, however, the teacher returns to work but continues to perform badly, you may embark on a competency procedure, giving her the chance to improve against pre-set targets and providing appropriate professional support. Failure to meet the required standard, after due time and warnings, could lead to dismissal.
Good record-keeping is vital. Should the governors' decision be challenged, the log of all your efforts on their behalf to achieve a satisfactory resolution is the evidence which would be needed to satisfy an industrial tribunal that the employee had not been denied her legal rights and had been treated properly and fairly.