You should be able to get the bank to tell you if the money order was actually cashed. If you can do this, then that means it was received and you can show your proof to the landlady. That would make her liable for the cost of your obtaining proof. But, to collect, you would have to sue in small claims court -- unless the landlady voluntarily pays. And, that would be a giant pain.
It's not fair, for sure, but you have the obligation to pay rent, and if you don't pay in full, the landlady can commence a summary action for possession/eviction. Your only protection is to pay the rent in full, and then sue to recover the cost of proving that you paid the rent timely.
In the future, you could use priorty mail, so that you have a tracking number. But, that costs money, too. If it were me, I would use a checking account with a bill payment feature, so that you have proof positive that your checks are reaching their destination -- and if they don't, then the bank will have to resend at its expense.
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