In reading a few laws recently, I've seen a lot of non specific words like "reasonably", "expeditiously", and "sufficient". I'm wondering how judges decide what is "reasonable" when the way the law is written is so subjective .
Judges do not decide, jurors do (however, if a judge is the fact-finder, then the judge makes such a determination). The main input that the decision-maker gets is a jury instruction. In order to unify "reasonable doubt", "reasonable price", "reasonable delay" and so on, appeal is often made a mythical being, "the reasonable man", so reasonable force would be the degree of force the reasonable man would use in a given situation. I will draw from California criminal instructions ('cuz I have them) but similar instructions can be found across jurisdictions. For example, one instructions says "A natural and probable consequence is one that a reasonable person would know is likely to happen if nothing unusual intervenes". Or from a negligence instruction "A reasonable person would have known that acting in that way would create such a risk". More detailed appeal to The Reasonable Person is found in the justified homicide instruction:
Defendant’s belief must have been reasonable and (he/she) must have acted only because of that belief. The defendant is only entitled to use that amount of force that a reasonable person would believe is necessary in the same situation. If the defendant used more force than was reasonable, the [attempted] killing was not justified.
When deciding whether the defendant’s beliefs were reasonable, consider all the circumstances as they were known to and appeared to the defendant and consider what a reasonable person in a similar situation with similar knowledge would have believed. If the defendant’s beliefs were reasonable, the danger does not need to have actually existed.
The defendant’s belief that (he/she/ [or] someone else) was threatened may be reasonable even if (he/she) relied on information that was not true. However, the defendant must actually and reasonably have believed that the information was true.
There is no explanation of what it means to be "a reasonable person". Since nobody believes that they themselves are unreasonable, a simple and also wrong way of judging the matter is to subjectively judge whether you yourself would do the same thing, if you were in that situation.
Very often, instructions do not even bother to say what "reasonable" means, so (re interpreting expert testimony) "You may disregard any opinion that you find unbelievable, unreasonable, or unsupported by the evidence"; (re corpus delicti) "That other evidence may be slight and need only be enough to support a reasonable inference that a crime was committed"; "Also, before you may rely on circumstantial evidence to find the defendant guilty, you must be convinced that the only reasonable conclusion supported by the circumstantial evidence is that the defendant is guilty", "when considering circumstantial evidence, you must accept only reasonable conclusions and reject any that are unreasonable"
The closest the law has come to articulating an objective characterization of "reasonableness" is in "reasonable doubt" instructions. One characterization is in People v. Feldman, 71 N.E. 2d 433.
It is not a doubt based upon sympathy or a whim or prejudice or bias or a caprice, or a sentimentality, or upon a reluctance of a weak-kneed, timid, jellyfish of a juror who is seeking to avoid the performance of a disagreeable duty, namely, to convict another human being of the commission of a serious crime
A somewhat improved characterization is the Calcrim instruction
Proof beyond a reasonable doubt is proof that leaves you with an abiding conviction that the charge is true. The evidence need not eliminate all possible doubt because everything in life is open to some possible or imaginary doubt
There is room for improvement, but it points in the right direction.
On rare occasions, a law is written that actually includes a definition. The Gas Price Spike Act HR 3784 said
The term ‘reasonable profit’ means the amount determined by the Reasonable Profits Board to be a reasonable profit on the sale.
It is then up to the board to subjectively determine what that profit is. (BTW this did not become law).
It's actually very difficult to determine what reasoning judges use in those cases where they are the determiners of fact. They will likely call on their knowledge of law, asking "are these circumstances sufficiently like past circumstance A where the defendant was convicted, or more like B where the defendant was acquitted".
As your post surmises there is no hard-and-fast rules - I'd argue this is a key reason why which justices are appointed to the Supreme Court can be such a political issue.
A lot of the answer to this idea comes down to the "Reasonable Person" test - which is purely hypothetical - in effect, the judge would ask him/herself what would a reasonable person expect in this circumstance, and be guided by that. Unfortunately this can be somewhat subjective.
Judges may also look to previous related decisions to help them look at what other judges consider reasonable, sufficient and expeditious.