@FantasyLoverWhat everyone else said is valid, however there are some literary standards concerning such things (yes, I'm aware that few here give a damn about 'literary works', still, there are certain well-grounded and proven ways of handling these 'issues': 1) Always credit the original author, including a link to the existing story (when rewriting it, which again, isn't what you're doing), 2) In most cases, rather than 'completing' someone else's story, as you're doing, you'd instead take the story premise and write a completely original work, so you'd then own the copyright of your own efforts, while each author benefits from the additional views of each piece (i.e. the more rereads and the more new chapters, boosts both of your overall numbers (ratings, hits, views, etc.).
Yes, Lazeez does allow others to complete stories which have clearly been abandoned, just realize that it is a non-standard usage, with the corresponding legal copyright issues.
And I commend you, for not just 'taking over' the entire story, but just adding a new summary conclusion, wrapping up the unfinished, incomplete story threads. Thus, I have no reservations commended your approach. It also protects the copyright protections for each of you, since posting online grants you the same copyright protections that publishing does. Though if you'd just 'taken over' the work, distant relatives of the author could always launch a legal copyright suit, where you would be liable for (that's highly doubtful, yet it's still a possibility, so it's better covering your ass legally), though since it's 'not for profit', the worst they could do is issue a 'take down' notice to remove the story, so you're doubly covered here.
That said, once you have written the summary, you could easily write a subsequent sequel, under your own name, with the same stipulations (listing both works, for the same reasons) and since it's completely original, you're still covered. They could sue you, yet wouldn't stand to gain anything from it, so there's little reason to file ANY suit.
Which is why I keep urging authors to familiarize themselves with both copyright and plagiarism rules, as otherwise, you effectively have NO protection for any legal challenge.