Back in the day there were entry level professional jobs. My first job was one of these: working on a hotel reservation system for NCR. A lot of these jobs have been outsourced to off-shore developers or the on-shore equivalent, body shops which are expert at getting H1-B visas.
The jobs that remain have moved up the “skill ladder”. As you relate, many companies are not only looking for a degree, but also people who have coding experience.
There is also the issue of bias. Programming interviews often consist of hours of solving problems on a white board. I deeply disagree with this approach to hiring, but reality is as it is, not as I wish.
This widespread white board style of interview is confrontational. You have to stand up at the white board and think quickly on your feet. This may be a style that selects against women.
The difference between your son and your daughter is that he has been more obsessed with computers and software. Unfortunately most companies don’t give points for being a well rounded person. They want someone who can help solve their problems.
Having a math background, particularly in statistics, can be a real asset in some areas. These include data science and finance. If your daughter has not applied to fiance companies like Blackrock, I recommend doing so. A background in R or Matlab is useful for these jobs (or, horrors, Excel).
Just as the only way to become a good writer is to write, there is only one way to become a good software engineer: write software.
If there is an application that your daughter finds compelling, I recommend that she built it and open source the code on GitHub. In doing this, I recommend building expertise with Ruby on Rails or the Python environment Django. I’d recommend Grails, which I used to build nderground, but it is not as widely used (unfortunately).
If no project comes to mind, then I recommend looking around for some open source projects to take part in.
Another fruitful path might be attending Hackathons, Women Who Code and other “Meetups”. There are always people who have company ideas who are looking for people to help them with technology. Some caution should be exercised because some of these people take advantage of the people who help them with technology. I liken Hackathons to Tinder dating. Be careful who you sleep with.
Don’t give up. There is always a path forward.