A New Hire is Sick and Can No Longer Work. Help!
This does not mean that you cannot fire a newly hired individual who is not capable of performing the duties of the job. It does mean, however, that you must be prepared to prove, if necessary, that you could not reasonably accommodate the absences/part-time attendance precipitated by her malady.
Many businesses cannot accommodate extensive absenteeism -- and yours may be one of these. If so, you should simply call her in and let her know that this is the case, terminate her employment and wish her well.
It appears that you have genuinely tried to treat her fairly and give her a chance to recover to the degree whereby she could perform her job duties. On the other hand, I have seen individuals claim discrimination and/or sue employers who have been more than fair regarding treatment of them when they had a disability -- even when the disability had nothing whatsoever to do with the termination of employment.
So, you should be prepared to deal with that possibility. Offering her a severance package of a couple of weeks of pay in return for her signing a separation agreement that contains language protecting your company from any kind of legal action would be advisable.