Who gave the order to drop the second bomb on Nagasaki?

Let's first look at who gave the order to drop the first bomb, because that will enlighten some of the question.

There was no great deliberation or decision as to whether to use the atomic bomb. There were discussions about how to use it and on what targets. It was finalizing this list of targets that led to the finalizing of a strike order on July 25, 1945. To give some indication of how difficult the "who" is in this situation, let's run down this strike order:

The order was drafted by General Leslie Groves, who developed the bombs and their operational parameters.
The order itself however is an order from General Thomas Handy to General Carl Spaatz.
The order was shown to Truman as a form of consultation, not because anyone thought he needed to authorize or approve it. He did not object, which would have been his only intervention if he had wanted to make it.
The formal approval of the strike order prior to it being issued was given by the Secretary of War, Henry Stimson, in writing (in a telegram, to be specific).
And of course if we broke it down to the operational level it gets very complicated: Captain Paul Tibbets was in charge of the B-29 mission that dropped the bomb, flying under the auspices both of the Manhattan Project's local Tinian command (General Thomas Farrell) and the overall command of the 20th AF (Gen. Curtis LeMay).

So you can see that even saying "who gave the order to drop the first bomb on Hiroshima" is complicated.

One thing about that strike order — notice its second clause:

Additional bombs will be delivered on the above targets as soon as made ready by the project staff. Further instructions will be issued concerning targets other than those listed above.

It is mostly an order about a specific first bomb, but then opens the door to as many bombs as are available. So no further follow-up order was given, or needed, from Washington in order to authorize a second strike.

The plan on Tinian had always been to drop both bombs, within about a week of each other. Weather conditions changed the initial drop date from August 3 to August 6, and forecast weather had the local Manhattan Project representatives on Tinian move the drop date of the second from August 10 to August 9. They did this, so far as I can tell, without alerting anyone high up the command to the matter — because they had already been authorized to use their discretion and to deliver bombs "as soon as made ready."

So for the first bomb, Stimson and Truman were both kept aware of when it was to be dropped, with updates coming by the day. For the second, they both learned about it after the fact.

So who authorized the second bomb? If I had to pick just a single person, I'd probably go with Thomas Farrell, the highest local authority to the Manhattan Project on Tinian, and the one who approved the operational change of moving the date up on the second strike. But even that's kind of unsatisfying, because he was just going off of his interpretation of the first strike order, and if the weather hadn't been different, he probably wouldn't have felt the need to give any specific order/authorization on the second bomb at all.

These kinds of issues, as an aside, are part of what make tracing clear lines of authority on the atomic bombs very difficult. Most people assume there were clear lines of command, even clear chains of command. In reality things were blurry not only because the bomb was regarded as "special" and thus lived somewhat outside of the normal chain of command, but also because the secrecy of the operation meant that it was not going through all of the normal channels anyway. Historians tend to think of the use of the atomic bomb not as the product of any signal "decision" but as the product of a long train of choices that many people contributed to.

A good read on these kinds of issues is Michael Gordin's Five Days in August, which really picks apart the ways in which authority is diffused in the project, with special attention to the ways in which the local authorities on Tinian were more consequential than people realize.