When done properly, osmotic shock should release only the contents of the periplasm. So if you are getting DNA contamination, then you are getting cell lysis and not just release of the periplasmic contents. There is no DNA in the periplasm, even if a little lysis occurs there should not be enough to cause noticeable viscosity.

You can test by measuring the activity of some enzyme that should be cytoplasmic and see what percent of that enzyme activity is in your osmotic shock fluid, relative to the total (by measuring whole cell activity before you start). This will tell you how much cell lysis is occurring. Which enzyme depends on which E coli you are using, but LacZ is obviously an easy one if your strain is Lac+.

However to answer your question, DNase I will have low (but not zero) activity at 4oC. Depending upon the strain of E. coli you are using, if it is EndA+ (and not an endA mutant common to many molecular biology strains) then this is a periplasmic nuclease and if you add some Mg++ it will take care of any DNA problem. It will also be low activity so in either case you might want to wait overnight at 4C before processing further. But I think you can solve your problem more readily by optimizing your osmotic shock.

Thank you Dr Benedik for your answer. I assumed that I need to optimize the protocol for periplasmic extraction because that space should not contain this much DNA. While the solution is not visibly that much viscous, the flow rate through the Ni-NTA beads is very low.

For your information, the strain I'm using is BL21 (DE3).