Hi, D. -- if you search past postings here you will find these questions asked in the past.
We have flown a lot with our daughter since she was an infant, since we have family overseas.
Buy a seat. It is worth the expense. Reason one: Safety. In serious turbulence, you just cannot depend on being able to keep your arms around a child and prevent the child from flying out of your grip. You would not dream of letting your child ride in a car without a proper car seat, right? So don't let her ride in an aircraft without proper restraints either. Take her car seat but check with the airline well in advance that your particular seat fits their seats -- measure your seat in all dimensions. Seats will advertise that they are airline approved but that doesn't mean a lot -- talk to the airline. Someone posted that the airlines require you to keep small children buckled all the time -- I haven't been ordered to do that, but keep my daughter buckled all the time anyway for safety, and it's not a big deal. Airplane seatbelts can be made quite loose to go around a child who's sleeping in an odd position. And if she's in a car seat it's a moot point anyway -- she'll be secured as if in the car.
Reason two: Your sanity. You will arrive hot, sweaty, exhausted and frustrated after a six-hour flight with a child on your lap (even if your husband has her half the time --then both you and he will arrive sweaty and exhausted). In her own seat it will be easier to keep her entertained and you might get a better nap out of her.
A night flight that short, under 6 hours, will mean she gets little real sleep (unless she's a kid who really conks out easily). By the time she actually gets over all the noise, excitement, new environment, pressure in her ears, etc. she will be some time into the flight, and then will get a little sleep before being forced awake by the lights coming on, breakfast carts rattling down the aisles, etc. Overnight can work, but six hours is just short enough that she would not get "a full night's sleep." See if you can do a daytime flight that covers her nap time because a nap may be all she can get, if that.
Bring new things to distract her. A new toy, new small game, crayons and paper, etc. I am not a fan of using portable DVDs etc. to distract kids since they get used to that stimulation and don't learn to entertain themselves if it's done repeatedly, but if you need to use the DVD route, that's fine too.
Even though she's not an infant, have her drink during takeoff and landing to equalize pressure in her ears. That pressur can be very painful for little ones.
Please don't medicate her -- some parents say to use Benadryl to get kids to sleep but the medication itself includes warnings not to use it just to make kids drowsy, and it can have the opposite effect -- it totally winds up some kids and makes them more awake! I didn't use it for flying but the one time I used it for my daughter for allergies, it wired her instead of making her drowsy.