The Sixth Amendment guarantees the right to a speedy, public trial in criminal prosecutions, protecting defendants from excessive, prejudicial delays between indictment or arrest and trial.
Sure but the courts have ruled that a "speedy trial" can take years (I don't agree with this, it's just what the courts have decided). Additionally if you're not a US citizen you don't necessarily have all the same rights as a citizen, and your case is processed in civil court. Anyone can be detained, and a good judge won't release illegal immigrants from detention since they're just not going to return to court until after their hearing.