Infection should have been obvious by then,
Unless it was a recent infection. When Harry ran away, HWWB found him very quickly. What if Justin learned HWWB was in the area and then found out Elaine had skipped school and was acting strangely? Justin concludes her strange behavior is the result of infection (and he could be right) and conducts an enthrallment ritual to cleanse her. In GS, Harry and Molly have practice mind-wars to train in mental defense magic. It follows that Nemesis infection, which seems very similar to a form of enthrallment, might be countered by actual enthrallment magic.
When Harry gets home, Justin wants to test him too. Again, because Justin recently learned that an Outsider recently arrived in the area. When Harry runs away, that confirms (in Justin's mind) that Harry must have been recently infected. When Harry comes back, Justin tries to cleanse him. When Harry fights off the enthrallment, Justin decides to open a gate to the NN, push Elaine through where he knows someone from Summer will be there to help her, and then either died before he could escape himself or took advantage of Harry's attack to commit non-permanent suicide.
I can think of several reasons why Justin decided to let Harry win after Harry resisted. First, Harry resisting the enthrallment proved he wasn't infected. Thus, Justin knew Harry was an innocent and didn't want to kill him but lacked the energy to incapacitate him again without hurting Harry. Second, with the house burning down and the heavy use of magic, Justin knew he wouldn't be able to subdue Harry, enthrall him to cleanse him and then hide Elaine and Harry without the White Council finding out. Or third, Justin was surprised by the display of power from Harry and was concerned Harry's new power might be from a deal with Outsiders. Thus, Justin couldn't be certain he could defeat Harry and focused his efforts on saving Elaine.
How would enthralling someone test them for Nemesis? In GS, we learn that Harry has been training Molly in mind magic in unsanctioned ways to defend against mental invasion magic. The way Nemfection works seems highly related to enthrallment, so it seems likely that enthrallment magic would be the most effective way to discover or reverse Nemfection.
If the above is true, then why did Justin wait so long? Well, enthrallment magic isn't benign and likely has the potential for permanent damage. Perhaps it is worse on young people so Justin wanted to avoid it as long as possible. Maybe Justin didn't ask Elaine to stay home from school. Maybe she did that on her own. Justin found the behavior suspicious and having also seen evidence there were Outsiders in the area, had reason to believe (perhaps correctly) that Elaine had been recently infected. When Justin tried to explain, Harry ran away instead of listening. When Harry came back, Justin thought that Harry had also been infected and tried to enthrall/cleanse Harry. When Harry resisted the enthrallment, maybe that was sufficient proof he wasn't infected. Having discovered Harry wasn't infected, Justin didn't want to kill him, but as Harry provided no opportunity to explain, the best Justin could do is push Elaine into the NN where he knew someone from Summer would be waiting to help her and then took advantage of Harry's attack to commit non-permanent suicide.