Complex sentence: In August 2015, Derbyshire announced on Twitter that she had been diagnosed with breast cancer and would be having a mastectomy, but would continue to present her programme as much as possible during treatment.
Split into simpler sentences: In August 2015, Derbyshire announced on Twitter that she had been diagnosed with breast cancer, and would be having a mastectomy. But would continue to present her programme as much as possible during treatment.