How much notice do I need to give my landlord before vacating my apartment on a month-by-month lease?

I currently rent an apartment with my wife in Los Angeles. We signed a lease when we first moved in, but this only lasted for a year. After one year the lease reverted to a month-by-month basis without us signing any additional doentation. My landlord has been giving us grief about repairs (blaming us for acts of God, calling and verbally haring me, etc.) and we've had enough. Starting tomorrow, March 25, we are going to start looking for a new place. My question is, say we find a place in the next few days, how much notice do we need to give my current landlord? The lease I initially signed reads as follows: "Renter shall give Owner 30-days written notice of intent to vacate the premises at the end of the lease term, at least 30 days before the end of the term and Renter shall be liable for payment of rent through the later of: the end of the term, the expiration of the 30-day notice or the date of vacating the premises." Does this mean apply even though I am paying month-by-month? Or will I have to pay for April regardless? I realize I am doing my landlord a discourtesy by moving out so soon, but in my mind she has been discourteous to us on several occasions. Help! Advice!