I have been trying to find a cheap airfare to Wichita, KS from LAX, but the prices are about $350. Is there anywhere else to fly into that isn't that far away and would be cheaper?

No. Try changing your travel days to fly on Monday thru Thursday in each direction. Tue + Wed might be best.

By looking at mapquest, it appears that the nearest major cities are Tulsa, OK and Kansas City, MO. Both of those are served by SW Airlines, but they are appx. 200 miles away each. I don't know if the cost savings justifies driving 200 miles. It wouldn't for me. I don't know what time of year you are travelling, but I grew up in Northern Oklahoma and there's always chance in the winter that the roads will be icy. As you are from LA, you probably aren't used to icy roads, so I wouldn't try it. Good luck, whatever you decide.

Would you consider Priceline?

Try AA into Long Beach or Orange Co.
Long Beach is literally a 15 minute drive from LAX and Orange Co is a little further south but might be worth it.
I think 350 is as good as you will get on a major carrier before Christmas (you don't mention dates)
Wichita is not a market with tons of competition.
If you need it significantly lower, you'll have to do Priceline.