、。
How long will you have been working here by the end of next year?
Literal
Next-year end [until-までで], here [at-で] work-began-from how-much-[possessive-の] period [become-に-なります] [question-か].
A complex temporal question. 来年末 ('end of next year') + までで ('by / as of') sets a future reference point. 始めて is the te-form of 始める here functioning sequentially. どれくらい+の+期間 ('how much of a period'). The whole sentence asks about cumulative duration up to a future point — Japanese has no perfect tense, so it builds this by stacking temporal markers.