、。
She forgot to bring her umbrella, so I lent her mine.
Literal
She [topic-は] her own umbrella [object-を] forgot-came [because-ので], my one [object-を] lent-and-gave.
忘れてくる ('to forget and come back / to leave behind') uses ~てくる ('come') in its abstract sense — a past action whose result is brought back to the speaker's location, i.e., she left the umbrella elsewhere and came without it. 私のを ('mine, my one') uses possessive の standing alone for 'mine' (literally 'my (one)'). 貸してやった is 貸す + ~てやる, the casual/condescending benefactive — 'I lent her [something].' Note this form would be reserved for contexts where やる feels appropriate, often with subordinates or close peers.