In response to the issue with memorizing multiplication tables, working memory could be the problem, but it kind of depends. It is hard to say, even with extensive testing, whether the problem is an encoding issue or a retrieval issue. Since she is trying basically memorize rote information that has no meaning to her, she is unable to integrate it into her long-term memory in any meaningful way. OR once it is in her long-term memory, she is unable to retrieve it because there are not enough cues for her to be able to access it. There is really no way to know this from a parent's perspective.
Rehearsal is one way of keeping something in working memory continually. Low working memory may mean that she is unable to rehearse the information in WM long enough to properly encode it into long-term memory
There was a famous paper on working memory (probably the most famous cognitive psych paper on working memory) about Miller's magical number 7. The gist is that people can keep about 7 + or - 2 (so 5-9 items) items in working memory at a time. That can be individual items, or chunks. For experts, it might be very large chunks of information. For kids/people with low working memory it might be 5 chunks/items or fewer. This can either be new information from the environment or information retrieved from long-term memory.
As far as I know there is really no way to increase working memory capacity. People can learn how to chunk information more efficiently which can effectively increase capacity, but nor really. It is still limited to that 5-7 pieces of information at any one time.
I have found that my son, who can remember anything that is read to him that has some sort of meaning, can't memorize things like multiplication tables either. I used that program that gives each multiplication problem a story. DS is so much better at remembering the story that I am. It worked really well for him. Now he just has to remember which items have stories and which don't because there are only story for about half of the problems. I really think that he is just about to encode and integrate information efficiently and then he is able to retrieve it very effectively. However, if there are not enough relevant cues for encoding and retrieving, as in multiplication tables, the information is lost.
Michelle