There is quite a big entry for him in the Dictionary of National Biography. I suspect I'm not allowed to quote this in full (copyright) but it's accessible at public libraries. It states
As with many from East India Company service overseas, Horsburgh became an old man with young children. At forty-three he married Elizabeth Longworth (d. 1829) on 14 October 1805, and they settled first at 6 Savile Row, Walworth. Their first daughter, Jane Frances, was born on 29 August 1808; a second daughter, Elizabeth, followed in 1815 and a son, James, in 1821 when Horsburgh himself was fifty-nine.
It also refers among its sources to these, all of which might be worth investigating:
‘Biographical memoir of James Horsburgh, esq., hydrographer to the East India Company’, Naval Chronicle, 28 (1812), 441–51 · GM, 2nd ser., 6 (1836), 98–9 · Journal of the Royal Geographical Society, 7 (1837), vi · A. Day, The admiralty hydrographic service, 1795–1919 (1967) · H. R. Mill, The record of the Royal Geographical Society, 1830–1930 (1930) ·