How on earth could it work?
As I already told you, there is simply too little time and too little genetic diversity.
Excessive interbreeding between extremely small and highly genetically similar populations is almost guaranteed to result in tremendous physical abnormalities after only a couple of generations.
There is a reason that extremely small populations die off, and it's because there is a need for genetic diversity. When you pair together highly similar individual genomes over repeated generations, the chances of passing on deleterious recessive alleles increases exponentially with each consecutive generation.
As I said, one human female producing a couple of children at most is simply way too much of a population bottleneck for this to work.
Read up on this subject some more and you'll begin to understand why her plan is a complete waste of time:
en.wikipedia.org/wiki/Inbreeding_depression
en.wikipedia.org/wiki/Population_bottleneck
en.wikipedia.org/wiki/Inbreeding
This wouldn't work either. She may be able to keep said sperm alive long enough for her own use, for for her children's use? Not gonna happen. The longer sperm is frozen the less chance there is that it will survive the thawing process. Not to mention we've already established that the majority of buildings around the world will lose access to electricity, and that includes sperm banks.
In the absolute BEST scenario, maybe one or two generations would be able to utilize sperm from a sperm bank, but after that they'd have to start breeding between each other, and there still would likely not be enough humans to make that work, and even if there was, they'd all be highly genetically similar, decreasing the chances that they'd be able to produce more than one or two generations, at best, of very many biologically fit offspring.
In short, there are simply too many factors conspiring against this plan.