If you're expecting to have to carry that word "hear" again you can relax. Here, the speaker simply states a fact, as a sort of addendum to his little bit about the chimney sweeping stuff.
Now he tells us that there's a "hapless" (i.e., unfortunate) soldier, whose "sigh / Runs in blood down Palace walls." Well, cool.
We didn't know sighs could actually run down walls in the form of blood.
You didn't know that because it doesn't really happen. This is all part of a gnarly metaphor.
The basic idea is that the Palace, which is here a symbol for government, royalty, etc., has blood on its hands, so to speak.
Okay, but what are we to do with these bizarre lines? Think of it like this. First, the soldier sighs about something (his recent wartime experiences, his government's military policy, etc.).
This sigh, an exhalation of breath, is the expression of whatever is bothering or upsetting the soldier. And we know that, because he's "hapless," he's helpless to do anything about what's bothering him—except, you know, sigh in blood.
The sigh runs in blood because, well, it has to do with the palace—i.e. the government that dictates policy in the first place.
It's like the soldier exhales, an ineffectual, "hapless" gesture that shows how powerless he is to change his situation. Instead, all he can do is defend the all-powerful Palace, or (worse) enforce its orders with violence (after all, soldiers tend to be trained to do that sort of thing).
And so, the expression of both his discontent and powerlessness (the sigh) turns to blood and runs down the palace walls. The Palace is marked by the bloodshed that the solider would be forced to carry out. This image, then, is another reminder of the "manacles" the speaker mentions in line 8. He says that these restrictions are everywhere, but now in this stanza he's giving us two examples to prove his point: the suffering chimney sweep and the solider—who as a tool of the "Palace" (or government), is powerless to prevent himself from causing the suffering ("blood") of others.