What is the name of the grammatical device where 18th and 19th Century sentences started with 'which'?

Solution 1:

I don't think one can meaningfully ask what is the name of this "grammar"? But expanding on my earlier comments, Killick is using which as a "discourse marker" (aka "filler").

It just so happens we don't use that particular word in that way today - but many of us might use alternatives such as er, um, ah, well, so, right, okay in OP's context.

I found these interesting observations in The semantic status of discourse markers (1997)...

... although some markers may be syntactically integrated, they equally often seem to be extra-clausal, and positionally variable, elements which make no predictions about, nor are they predicted by, the presence of other syntactic elements in their host units. Their function is typically adverbial or interjective, and as such they may be deleted from their host units without this resulting in ungrammaticality.

That's in the context of the author discussing the grammaticalization of discourse markers, by which he means the process whereby some given term starts out as a "content word" (a noun, verb, etc. which does actually reference/mean something specific), but gradually becomes a "function word", defined on Wikipedia as...

Function words are words that have little lexical meaning or have ambiguous meaning, but instead serve to express grammatical relationships with other words within a sentence, or specify the attitude or mood of the speaker.

TL;DR: In OP's examples, initial Which (which as a "content word" would be a pronominal reference to something explicitly presented earlier in the text/discourse) is effectively a "function word" with little or no semantic content (about equivalent to initial Well as used today).

Solution 2:

The Oxford English Dictionary mentions this usage as irregular / pleonastic, and hence spread into vulgar usage, with examples.

**** Peculiar constructions. (See also senses 7d, 8c) 14.

a. (as pron. or adj.) With pleonastic personal pronoun or equivalent in the latter part of the relative clause, referring to the antecedent, which thus serving merely to link the clauses together: (a) with the personal pronoun (or the antecedent noun repeated) as subject or object to a verb (principal or subordinate) in the relative clause, which is usually complex; (b) with genitive of personal pronoun (or equivalent, as thereof), which together with this being equivalent to the genitive of the relative (whose, of which): cf. that pron.2 9.

(a) c1374 Chaucer Troilus & Criseyde ii. 654 Þis is he, which þat myn vncle swereth he mot be ded. 1449 R. Wenyngton in Paston Lett. & Papers (2005) III. 68 Yowre wurschupfull astate, the whyche all myghte God mayntayne hyt. a1525 (▸1481) Coventry Leet Bk. (1908) II. 493 Which yf it so be, we haue gret cause of displeasure. 1526 Bible (Tyndale) John xxi. f. cliij There are also many other thynges which Iesus did: the which yff they shulde be written every won, I suppose [etc.]. 1589 G. Puttenham Arte Eng. Poesie iii. iv. 122 Ye finde these words, penetrate, penetrable, indignitie, which I cannot see how we may spare them. 1655 T. Fuller Church-hist. Brit. ix. 175 A Schedule containing his heresies, (which what they were may be collected by that which ensueth). 1690 J. Locke Two Treat. Govt. (1694) ii. v. §42. 196 Provisions..which how much they exceed the other in value,..he will then see. 1726 G. Shelvocke tr. Imperial Comm. in Voy. round World Pref. p. vii Scandalous and unjust Aspersions..which, how far I deserve them, I shall leave to the candid opinion of every unprejudiced Reader. 1768 L. Sterne Sentimental Journey II. 140 The history of myself, which, I could not die in peace unless I left it as a legacy to the world. (b) c1374 Chaucer Troilus & Criseyde ii. 318 Þe kynges dere sone,..which alwey for to do wel is his wone. 1470–85 Malory Morte d'Arthur xvii. xi. 705 Ther is in this Castel a gentylwoman whiche we and this castel is hers. a1533 Ld. Berners tr. Arthur of Brytayn (?1560) lxv. sig. Piiiiv To do many thynges the whyche the hurte therof lyghteth on theyr owne neckes. 1622 J. Mabbe tr. M. Alemán Rogue ii. 164 Take away..mens credits, and estates.., which lies not afterwards in their power to make restitution thereof. 1721 R. Bradley Philos. Acct. Wks. Nature 90 Bulbous-rooted Plants, which when the Leaves of them decay, a new framed Root..supplies their Loss.

b. Hence, in vulgar use, without any antecedent, as a mere connective or introductory particle.

1723 Swift Mary the Cook-maid's Let. 13 Which, and I am sure I have been his servant four years since October, And he never call'd me worse than sweetheart, drunk or sober. 1862 Thackeray Adventures of Philip xvi ‘That noble young fellow’, says my general... Which noble his conduct I own it has been. 1870 B. Harte Truthful James, Answ. to Let. viii Which I have a small favor to ask you, As concerns a bull-pup, which the same,—If the duty would not overtask you,—You would please to procure for me, game. 1905 Daily Chron. 21 Oct. 4/7 If anything 'appens to you—which God be between you and 'arm—I'll look after the kids.