DNA Appliance or jaw surgery? (Photo)

The facial asymmetry will respond well to the DNA Appliance.  If the Cant does not correct I would suggest than doing cranial work (chirodontics) and be sure to work with a neuromuscular dentist.  A simple cant can sometimes be almost magically corrected by a combination of neuromuscular diagnostic orthotic and a Bob Walker trained chiropractor.  
There are many methods of doing maxillary expansion, some are primarily dental while others are orthopedic.  If there are braces and brackets chances are you are primarily moving teeth.
In children Rapid Maxillary Expansion is amazing but by 12 years old that window is closed.
For over 20 years my first choice was Rapid Maxillary Expansion with a surgical assist, now called the SARPE Procedure.  It is still an excellent procedure and offers rapid expansion.  This is faster than the DNA Appliance.
Orthognathic Sugery is fastest approach but carries considerably more surgical risk!

Non-surcical maxillary expansion can be done with a functional appliance like a Schwartz or Crozat, again something I used to employ but there are now better treatments in my opinion.
My first Choice is the DNA Appliance and Epigenetic Orthopdeics/ orthodontics and the mRNA version for sleep apnea/ snoring patients and patients who need advancement of the chin to create a better profile and airway.  Best results but requires some patiences.  Ideally appliance is worn 16 hours/day.  Less wear time equals longer treatment times.
I also utilize the ALF Appliance but feel I get the best maxillary  expansion with the DNA Appliance.  For a patient like you I might consider initial DNA and then switching to ALF but that increases total cost compared to just using DNA.
These can be used in combination starting with DNA and then an ALF for the next stage.  The ALF is very comfortable and works well with OMT
Regardless of the technique consider working with an Oral Myofunctional Therapist (OMT) for best results.
Ira L Shapira DDS, D,ABDSM, D,AAPM, FICCMO
Past Chair, Alliance of TMD Organizations
Diplomat,  Academy of Integrative Pain Management
Diplomate, American Board of Dental Sleep Medicine
Regent & Fellow, International College of CranioMandibular Orthopedics
Board Eligible, American Academy of CranioFacial Pain
Dental Section Editor, Sleep & Health Journal Member Cranio Editorial Board, Journal of Craniomandibular and Sleep Practice
Member, American Equilibration Society
Member, Academy of Applied Myofunctional SciencesMember, 
Academy of Cosmetic Dentistry Life Member, American Dental Association