They go back to the past and return to the two agents and Jeff & Arnau on the shore. Because government witnesses and journalists are there, the machine's existence is made public. Kenneth is cleared on his charges and gets the patent. Everyone lives happily ever after.
Which is why they never came back immediately after leaving. They went into an alternate timeline and thus came back to present in said timeline.
...a show called TIME BOAT. Kenneth and Darius are stuck in a Sliders kind of concept. (yes, I know it's not really but admit it, it SOUNDS like a TV Show!)
That's why it's built into a boat - there's a heat sink underneath to transfer the extra heat to the water.
He has no story at all. He wasn't shown documenting anything and it's going to be difficult to get any proof of what happened. He'll probably just look like a madman.
- If he leaves out the machine actually apparently working (and probably the government agents actually being there), it'll just be a human interest story about an eccentric kook. Even if it doesn't get run (not interesting enough, the government agents squashing it, whatever), his editor wasn't expecting a story about a real working time machine.
- He may just be confused about people finding him in person when the ad only gave a PO box. It's entirely possible that he indeed did place the ad after he and Darius made their trip, but he probably would have somehow informed or otherwise prepared his past self about it.
At the end, we just saw it jump out, but whose to say it didn't immediately atomize itself, and end the story right there? (Downer Ending or not, this is actually the tidiest explanation for the whole thing, leaving no paradoxes, ambiguities, or violations of causality.)
The first time Kenneth went back in time he did it alone. He saved Belinda but then, realizing than she was never interested in him, ended up crashing the car in her living room. The time travel messed up his memories. That's why he took a moment to react when Darius tells him she spoke with Belinda—one part of him remembers the timeline where she dies and another the one where she lives.
Kenneth isn't shown to be delusional about anything else (his machine does something and goverment agents are following him, though not for the reason he'd assumed), so it's unlikely that he's deluded himself about Belinda. So Kenneth is right - he remembers the original timeline (in which he did date Belinda and she died) even though he's living in the one he will/did alter (in which whatever he had to do to save her made him just a weird friend with a crush instead of having dated her). Similarly, Darius's mother might be alive, but she remembers her dying since she hasn't changed that yet.
Anything that could cause a grandfather paradox is just covered with whatever happened in the original timeline. When Kenneth or Darius are thinking about using time travel to save Belinda or her mother, that's the original timeline coming through. Any other time, their memories reflect the current timeline. Once they complete their respective missions, they'll always remember the current timeline.