I am a lodger & I dont have a tenancy agreement or contract. How much notice should I give?

Do you pay rent every week or every month? if it's every week, you only have to give a week's notice. If monthly, a month's notice. The only way you have to give 60 days is if you pay rent every 60 days. Does she hold a security deposit for you? if she does, I wouldn't py the last month/week's rent - let her keep the deposit in lieu.