Flight instructor with student reported a NMAC with a solo student when the solo student did not follow ATC pattern instructions. Instructor stated that the solo student then landed off runway.

Narrative

I was flying with my student in a Cessna 172 at ZZZ. We were conducting training touch and go's for about an hour. A solo student in Aircraft Y was conducting her solo behind me in sequence. My student made a touch and go on Runway XX. On the go; Tower instructed us to fly runway heading. On the climbout; my student asked me to take the controls so I became pilot flying. Tower instructed me to make left traffic so I turned left crosswind at pattern altitude. Aircraft Y took off behind me; Tower instructed her to fly runway heading. Upon reaching pattern altitude on climbout; Aircraft Y made left traffic despite being told to fly runway heading. Aircraft Y cut me off in the pattern while I was on downwind. I believe we missed hitting each other by about 500 feet horizontally; we were both at pattern altitude. I noticed this and began a right 360 for spacing. As I started this maneuver; Tower instructed me to make a right 360. I'm not sure Aircraft Y realized their mistake at this time. Once I completed my 360 I rejoined the left downwind behind Aircraft Y and gave the controls back to my student. Aircraft Y was cleared to land first on [Runway] XX; then I was cleared for the option #2 to follow Aircraft Y on [Runway] XX.As my student turned final I noticed that Aircraft Y was not aligned with centerline on final. I was wondering why she wasn't making a go around; I thought she was still flying. Then I noticed she was stationary. As I'm realizing this; I hear Tower ask Aircraft Y if she's okay to which she replies that she is okay but she is in the grass. Tower instructs us to go around; to which my student flying complies. I noticed the student came very close to hitting the ILS building to the left of Runway XX. I think this situation could have been avoided if the solo student had more training. The wind was 190 at 5 [Kts]; we were using Runway XX; perhaps the student overestimated the wind correction and veered off to the left side of the runway. I think that Tower should have also told the solo student the reason why she had to fly runway heading (she had to fly runway heading to follow me in the pattern). Overall I think Tower and Ground handled the situation professionally.