RMedina - hi, it sounds like the chain that connects the handle to the flapper valve has too much slack in it. When you flush, the handle and chain's not pulling the flapper up high enough to allow the tank to empty completely before the water movement forces the flapper to seal again.
Turn off the water to the tank, take the cover off the tank, then empty the tank by flushing. Watch and make sure the handle, chain, and flapper aren't hitting other components inside the tank. Adjust the length of the chain so there's as little slack in it as possible when the flapper is closed. When you push the handle, the flapper should raise to a 75-degree angle or so (but shouldn't go more than 90 degrees). Once you're done, turn the water to the tank back on and test it by flushing.
If this doesn't seem to work, you may need to swing by a H-ome D-epot or L-owe's, visit the Plumbing section, look at the displays and talk with someone there. They may be able to locate the same repair kit your landlord used, and help you determine what other corrective actions you can try.
I'm not a plumber, but merely a DIYer who's installed some of those repair kits in the past. They can be installed wrong but the problem you're describing sounds like a fairly easy flapper chain length adjustment is all that's needed. Let us know if this helps you or not. Regards! Jim D/West Point, VA