Some of the replies seem to think you can make a 13 mile commute in your work clothes.
Perhaps you can if your ride is very flat and you are in no big hurry to get there.
Mine isn't and riding a bike is an aerobic exercise thing for me. If I ain't sweating, I ain't doing it right. My ride is maybe a half mile shorter than yours.
By no means is this a recommendation to buy cycle specific clothing. The only cycle specific clothing I have ever bought is cycling shorts and tights, shoes and gloves. I buy these because I think the difference between cycling shorts and regular shorts is well worth it. I never have and likely never will buy a cycling jersey. Generic poly athletic shirts work every bit as well. The local salvation army/good will has racks full of them for a dollar or two a piece. You could ride in cotton as well. It's a personal preference although cotton as a base layer in cold weather is a really bad idea, IMO. I have ridden in weather down to the low 20s with this sort of stuff. A long sleeve poly and a fleece or 2 and maybe a windbreaker is sufficient. I have winter tights but my first 2 seasons, sweatpants over cycling shorts did fine. Just find tight fitting sweats.
Clipless shoes/pedals are also a worthwhile expense. Cages work very well, if you are tightly strapped in, but, if you are commuting in traffic where quick unplanned stops are a regular occurrence, being strapped in tight can suck....badly. An advantage to cages is that you can wear boots with them in cold weather. I bought a pair of neoprene booties instead. They can be found on sale on line for under 20 bucks.
13 miles on a mtb is tough enough already. doing it on regular platform pedals will make it a lot tougher. find the money for clipless. i recommend spd specialized tahoes. they are easy to walk in, stand up well and are cheap.
for lighting, it depends. do you need lights for seeing or to be seen? cheap led flashing lights will do you for "be seen" lighting. you can get a pb superflash tail light/blaze headlight for under 50 bucks. for "see" lighting, the magicshine seems to be very popular and is under 100 bucks.
much of these things can be had from CL or ebay with pretty nice discounts from new. the one thing I'd be hesitant about buying used is shoes, but, I'd buy them as well if the price was right.