Institutionalisation, unless privately paid for, would seem an unlikely option as national systems of 'lunatic' asylums were established only in the 19th century in most countries. In most of Europe, suicide and attempted suicide were criminal offences until around 1790-1820; and much later in England and Russia. In Russia, until 1835, attempted suicide was punishable by death and this was followed by bodily desecration and profane burial.

For obvious reasons, suicide, rather than attempted suicide, has received more scholarly attention. For the 18th century the general argument is that suicide became gradually secularised and - partly out of sympathy for surviving family members as the property of suicides could be seized - juries increasingly returned verdicts of non-compus mentis. However, more recently scholars have argued that the medicalisation of suicide did not simply equate to the disappearance of older moral frameworks - but rather through a process of hybridization both approaches coexisted.

Of course, it's hardly likely that all suicide attempts resulted in arrest. For instance in 1734:

’On Thursday last [i.e. 23 May] a poor clergyman of the Church of England attempted to hang himself on a tree in a ditch near the Bridge at Little Chelsea; but was happily prevented by William Draycott of Great Chelsea, Esq: who chanced to be riding by at the Time, when he was putting the Rope about his Neck’ [...] ’However he drowned himself the same Night in the Thames near Fulham.

Michael MacDonald (1986) The Secularization of Suicide in England 1660-1800.Past & Present (111): 50-100.

DM Shepherd and BM Barraclough (1997) Suicide - a traveller’s tale: A study of the adoption of the word ’suicide’ into the main romance languages. History of Psychiatry 8(3) History of Psychiatry 1997 8: 395-406

Róisín Healy (2006) Suicide in Early Modern and Modern Europe. The Historical Journal 49(3): 903-919

Susan Morrissey (2004) In the Name of Freedom: Suicide, Serfdom, and Autocracy in Russia. SEER 82(2): 268-291.