Johnny Depp has been keeping a bedside vigil for his young daughter since she was hospitalized in London last week. Lily-Rose, 7, is suffering from an undisclosed illness. Johnny, 43, and his longtime girlfriend Vanessa Paradis, 34, are also parents to son Jack, 4. We are happy to report that their daughter is doing much better, Depp’s rep said Thursday. The family greatly appreciates the continued support and respect of their privacy.When Lily-Rose fell ill, Johnny was filming a movie version of the musical ‘Sweeney Todd,’ but the schedule has been adjusted to accommodate his needs.