As additional information to @livinginchina's list.
Ref. item 2. If you wife does not have a Kunming Hukou she will also need to bring her Kunming temporary ID and copies of that.
Ref. item 4. You need to get the passport photos from an authorized photo office, these produce a receipt with a bar code and your photo on the paper. This barcode is read by the PSB system. The photo office outside the Tuodong Lu building does this. The photo office for foreigners is in the outside wall, on the right hand side as you face the building, not the one on the corner of the building, it is about 10m further back.
Additional item. Your wife will need to write a letter of invitation, there is a standard form for this available in the visa office.
Things are still up in the air. However, I would speculate that teaching or tutoring school age children will be restricted, perhaps to the point of near impossibility.
However, teaching university students and adults is still permitted. Universities do not pay the best, but the contract is clear cut, there are usually no office hours (you teach 16 class hours max and the rest of the time is your own) accommodation is usually provided (sometimes an accommodation allowance) and flights home are paid. For those who have never worked in China before, universities are a good place to start and learn the ropes.
Adult learners are most common in the tier one cities like Shanghai, Beijing, Shenzhen, etc. When I lived in Shanghai I worked for 3 different companies over 3 years and I only taught adults.
@livinginchina, there are all those English teachers and other subject teachers already in China, who can easily have work permits renewed. There has been a trend over the last 10 years for more 'schools' to offer extra-curricular activities, mainly sports, which is attractive to both the parents of prospective students, and for some teachers. I am sure that 4 hours of paid sports coaching as part of any teaching contract would be attractive to many potential staff.