Broken component capacitor or resistor near battery wires

Sorry, But without the schematics I cant help much, But just know that type of part it would be better off either purchasing a new one. or you could fix it with the following part.

iPod touch (Gen 1) 8 GB Logic Board and Battery

But it costs the same as purchasing a new one (almost as much).

That missing part is so small that only a machine or a highly trained 2M Solder repair technician could repair it.

If you had the part (its small and hard to pick up with fingers.) i would use transparent tape to place it in the area of the trace and use a heat gun to reset it back in the spot. But even then the air pressure could blow it off the board if the transparent tape melts too fast before the solder melts and it sets. It is way to hard to repair with out the proper equipment and skill level. Better off getting it repaired with a new logic board or replacing the unit .