The Crown Prince is Frederick, Crown Prince of Prussia, son of Kaiser Wilhelm I, father of Kaiser Wilhelm II and married to the Crown Princess, Victoria (Princess Royal of England, daughter of Queen Victoria).
His illness referred to here is cancer of the throat and there is much controversy about the diagnosis and treatment - part of this is caused by friction between the eminent doctors concerned. The Crown Princess only has faith in an English doctor, Sir Morell Mackenzie, who is mentioned later in the letter; the German press believes that he is not as competent as German doctors. It is at about this that stage Sir Morell decides that the growth in the throat is malignant. (In June 1887 a Berlin Pathologist had found no sign of malignancy)
Early surgery is recommended by the German doctors - but Sir Morell wishes to delay this as much as possible.
Frederick's death in 1888 (after ruling as Emperor Frederick III for just 99 days) does little to clear this up - while the actual death may have been accelerated by a careless minor operation the death was primarily caused by a malignant cancer which may have developed after Sir Morell's diagnosis.
Prince Frederick and the Crown Princess are in San Remo in Italy at this time in the hope that he will be more comfortable resting. It is considered significant that the future Kaiser remained out of Prussia even though his father, Kaiser Wilhelm I, was approaching 90 and in very poor health.
Frederick's son Prince William (Ada's employer) is scheming to become Kaiser directly, bypassing his father on the grounds of his illness. (This probably explains the reference to secrecy a few lines further down in the text.)
NB Victoria (above) becomes the Empress Frederick in 1888 - and is frequently mentioned subsequently.