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Вот исправленные предложения:

• Tom asked me if I was free.
• I asked him why he was tired.
• We wanted to know if Kate had broken the car.
• He asked if they could meet on Sunday.
• I asked how often she used the washing machine.
• Mother wondered if we would invite Kate to the party.
• Tom said that he had invented a new computer program.
• Mary said that she would help her sister.
• They told that they had been really happy.
• She said that she lived a big flat then.
• Tom said that he was going to play tennis.
• Mary said that she wanted to watch TV that evening.
• Tom cold asked to keep silence. (quiet)
• Mother advised the kids to wash their hands before lunch.

В последнем предложении исправлено "hand" на "hands", чтобы соответствовало множественному числу, так как детей несколько.