There is no hard and fast rule for how long the state can take to indict a person, although there are statutes that require that bond be set after 90 days if he is still in jail and that set a statute of limitations (years) for initiating a prosecution. I hate to say it, but it's not all that unusual to have a person not indicted after 6 months, although it usually isn't that slow. Unfortunately, his public defender is right, there are not many things an attorney can do to force the issue until the case has been indicted.
The short answer is you do have to wait for the state to indict before you are able to file most substantive motions, including demands for speedy trial. The main exception to this is if the defendant is held in jail for an unreasonable amount of time without an indictment, in which case you may have a stronger argument that a bond should be set, or lowered, if a bond has already been set.
The longer answer is that the primary limitation on the time frame for an indictment is the statute of limitations relevant to the particular crime you are dealing with. However, depending on the length of time before the indictment and eventual trial there may also be an issue that one's Constitutional right to a speedy trial has been denied. For instance, if they delay of the indictment hurts your ability to investigate the case and present a defense, then you will have a stronger Constitutional speedy trial issue. Generally the statute of limitations for crimes in Georgia are found in O.C.G.A. section 17-3-1. I'm not familiar with any crime entitled failure to unregister, but presuming you are saying your husband was arrested for failure to register as a sex offender and presuming this is the first time he has been charged with the offense then the statute of limitations would be 4 years. So if they wait more than four years to indict they should not be able to proceed with the prosecution. In the mean time, you just need to consult with an attorney about specific steps that should be taken to prepare for your defense and maximize your chances of a good result in the case.