Landlord of the Dog and Duck
or Dr. Johnson as he hated to be known
I don't feel confident at attempting to type him but I will nonetheless make some summary that might help as a starting point. This is in large part from watching a documentary, reading his wiki page and other internet pages recently, and any lingering impressions I have from James Boswell's biography.
He is now widely believed to have had Tourette's syndrome which may well have had a significant impact on his interactions with others.
He seems to have been a somewhat abrasive sort of fellow in communication, perhaps not naturally gregarious. However, he was a very social individual. He enjoyed teaching in his early years (this involved teaching a small handful of students at a time) and regularly meeting with friends in public places etc. He basically pined when he wasn't in the company of others and maintained a network of acquaintances which only lessened in later life. He apparently spoke with confidence the king (George III) ("During the whole of the interview, Johnson talked to his Majesty with profound respect, but still in his firm manly manner, with a sonorous voice, and never in that subdued tone which is commonly used at the levee and in the drawing-room. After the King withdrew, Johnson shewed himself highly pleased with his Majesty's conversation and gracious behaviour. He said to Mr Barnard, 'Sir, they may talk of the King as they will; but he is the finest gentleman I have ever seen.").
He wrote several political works and was well-known for openly voicing his opinions. He is of course famous for his dictionary and also his biographies and literary criticism. Even his dictionary is famed for displaying his often strong opinions on subjects!
He had a pretty incredible work rate and always seems to have been working in one way or another. He does not seem to have been especially bothered for the condition of his house or the state of his numerous books.
He is less well-known for his poetry - I personally don't rate it, but apparently many people have done so. He hated writers and poets who wrote in overly flowery metaphoric or vulgar language and his own poetry is itself essentially functionally descriptive (which has been a criticism of his work and why I find it so lacking). He thought it important to categorise the English language, hence his dictionary, and thought it important that authors would be understood, hence his literary criticism was intended to be a practical exercise.
I don't want to unduly influence an attempt at typing but something in the range of LIE, SLE, LSI seems right to me. I am not at all sure how the writings of such types should come across in such an individual and from such a time - I only think of Martin Luther for example who has been typed as a SLE by some socionists and the language he used. Taking into account the types of people such as James Boswell might be important.