Wanting to buy used car, how to negotiate the price?

My husband and I are college students and I only have one year left. We have a Honda 2005, that seems like there is something always wrong with it. Before my husband and I got married, he bought the Honda, but his parents told him he could only get that car or they wouldn't co-sign. The car still has 3500 on its loan and probably another two years of payments. It also has 235,968 miles on it.
We just took it for an oil change yesterday and supposedly the transmission is leaking as well as the axel, but we don't see any leaks.
Yesterday, we went to look at used cars and found a Hyundai Elantra 2013 for $13,400 and it has 51,000 miles on it. We want to get a better car that will actually last us, instead of fall apart on us and we do not want to put more money out on it than it is actually worth. I know this is a rough situation and the car is probably not worth $3500. I mean it is probably in fair to good condition.
How can we negotiate for a better price than $13,400?

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