That summer in London was one of the strangest summers that Sherlock had experienced. Mary and John were more willing to have Sherlock babysit, considering that both kids had taken to Sherlock and he seemed to do a really good job with the responsibility.
It seemed as though Sherlock spent the summer in the park. Mary had taken a part-time job as a receptionist, and as result, Sherlock often had the kids during the morning and mid afternoon. Because Alex was an exuberant little boy who was always trying to find some way to get out of the confines of the flat and Isabel was starting to become more adventurous, Sherlock decided that he was going to get the kids out of the flat and let them get some fresh air.
As Sherlock found out, the park was a lovely place to sit back and figure out people's lives. Isabel, since she was still too little to walk, enjoyed the swings. Sherlock and Isabel spent a lot of time at the swings at the playground. This was a perfect situation for Sherlock—Alex would run around nearby, and Sherlock could scrutinize the other parents without seeming too out of place.
But, the park swings became dull after a while. Alex could spend hours on the play structure, but there was only so much time that the swings would remain interesting to Isabel, who was wont to fall asleep in the swings. As a solution, Sherlock decided that he was going to start teaching Alex and Isabel science, based on what they could find at the park.
Of course, the children were already behind on their basic chemistry education, but Alex and Isabel were intelligent enough to catch up. During the hours that they spent at the park, Sherlock would explain the finer details of virtually anything and everything that Alex pointed out. Alex learned about grass and its biological functions, chemical structures, and uses, not to mention the chemical compounds of the plastics used in the play structure. Isabel learned colors and shapes, and by the end of the summer, could identify different colors by nonverbal means.
In fact, Isabel's first word was "green".
The time that Sherlock had spent with the children on a regular basis had only been about three months, but during that time, the two kids became more aware of the world around them. Alex became more inquisitive, and Isabel learned to walk in the grassy field. Sherlock was convinced that Isabel learned to walk because she saw her older brother running around, examining everything with keen interest and she wanted to join in. John pointed out that she was old enough to learn how to walk, but Sherlock adamantly clung to his conclusion.
Whilst spending his mornings and early afternoons with the Watson children, Sherlock was still productive with ending the Moriarty crime web. Three more members were taken out that summer, leaving him with only five people to take out. It was a success, but Sherlock didn't seem as pleased as he should have been. His despondency lay with the fact that Irene and Adele were nowhere even remotely close to London and instead of spending the summer in the park with his daughter, he was forced to settle for the two Watson children.
It kept him going, knowing that maybe there was a chance in hell that he could relocate them to the Northern Hemisphere somewhere. Maybe there was a chance that Adele could study in London for university. But, that would be years away, and Sherlock couldn't think that far ahead without his head spinning, driving him to insanity. So, he tried not to focus on that, and instead, directed his efforts to less emotional means, trying to ignore Irene and Adele when he wasn't talking to Irene through video chat.
By the end of the summer, he began his countdown to the end of October. It was the day that he would fly to Darwin for Adele's birthday that kept him going. He had no idea when that had started, but the moment that he had acknowledged this fact, it became easier to accept that he actually missed them. He missed his Australian life as Paul Jenkins, and he missed his family—his daughter, his dog, and his partner/girlfriend/wife/parental counterpart/etc.