In Richard Horwood's 1819 map (extract reproduced here) the basin with Island are drawn to scale; nearby rows of houses are named 'Island Row' and 'Mill Place', respectively.
 Those streets still exist in Limehouse but the "Island" and the "Mill" themselves are largely forgotten.
 They originated as follows.

Charles Dingley was an entrepreneur and speculator who built, amongst many other things, the southern end of the Limehouse Cut.
At that time wooden planks were expensive because they were sawn by hand, or imported ready sawn; in England there was hostility to sawmills, which were thought to be illegal.
 Dingley, who was also a major timber merchant, decided to defy convention by building a wind-powered sawmill.
 He bought up a "stranglehold" of land in Limehouse and built the sawmill strategically close to the line of the intended Cut, which he may have promoted with that very intention.
 It was the only sawmill in England and did good business.

However on 10 May 1768—the day of the St George's Fields Massacre riots—a mob of 500 men, including hand-sawyers, attacked his sawmill and destroyed the machinery—anticipating the Luddite machine-smashers of later years.
It caused Parliament to pass an Act making it a felony to vandalise mills.
 Dingley repaired his mill by 1769.
Surviving records are consistent with the island and basin being constructed to isolate and protect the repaired sawmill.
In Richard Horwood's 1819 map (extract reproduced here) the basin with Island are drawn to scale; nearby rows of houses are named 'Island Row' and 'Mill Place', respectively.  Those streets still exist in Limehouse but the "Island" and the "Mill" themselves are largely forgotten.  They originated as follows.
Charles Dingley was an entrepreneur and speculator who built, amongst many other things, the southern end of the Limehouse Cut. At that time wooden planks were expensive because they were sawn by hand, or imported ready sawn; in England there was hostility to sawmills, which were thought to be illegal.  Dingley, who was also a major timber merchant, decided to defy convention by building a wind-powered sawmill.  He bought up a "stranglehold" of land in Limehouse and built the sawmill strategically close to the line of the intended Cut, which he may have promoted with that very intention.  It was the only sawmill in England and did good business.
However on 10 May 1768—the day of the St George's Fields Massacre riots—a mob of 500 men, including hand-sawyers, attacked his sawmill and destroyed the machinery—anticipating the Luddite machine-smashers of later years. It caused Parliament to pass an Act making it a felony to vandalise mills.  Dingley repaired his mill by 1769. Surviving records are consistent with the island and basin being constructed to isolate and protect the repaired sawmill.
Limehouse Cut