Landing a free agent is never easy. The Jets might have a few built in advantages over other teams in this area with Mathias Kiwanuka if they are interested, though. The first is that Kiwanuka's current team might make a big effort to bring him back. The Giants already have a lot of quality pass rushers on their defensive line, Justin Tuck, Osi Umenyiora, and Jason Pierre-Paul. They might not be willing to spend big on a part time player whose best qualities are strengths for the team anyway. General manager Jerry Reese has already indicated the team might be willing to make a one year offer. Avoiding a bidding war with a player's original team is always a plus.
I am sure that Kiwanuka will have suitors. Pass rushers are a hot commodity. His services might be relatively easier to contain than those of others, though, because he is coming off a pretty serious injury and has been a part time player through much of his career. When he has gotten a chance, Kiwanuka has been very good, but he does not bring a Pro Bowl resume to the table. He could thus end up being something of a bargain.
The Jets can offer him some great things. They can provide him the chance to be a full time player without having to move and uproot himself. Some suitors might not be able to give him the kind of playing time the Jets could. Nobody else can offer him the chance to stay where he currently lives on top of a ton of playing time. This wouldn't make a difference if another team was to blow the Jets out of the water financially, but it could make a big difference if things are close.