Often SS and other metals work harden due to dull cutting tools, Drills are the biggest culprit, I have found that reducing feed just before breakthrough greatly decreases that hard spotcaused by that last .030 of material.
If a piece is already got hard spots you need to have it normalized. Which means its put in a furnace heated to curie point around 1500, held for 30 minutes then slowly cooled down to 1000° or so at which point it then be air cooled. Different Temp and times for each alloy but thats the procedure.
The general idea is to avoid rubbing at all costs.
So for example if you are drilling a hole, lower the RPMs and feed aggressively. The idea being to make sure the cutting edge never rubs instead of cutting. You want the cutting edge to always be under what hardened on the previous revolution of the drill.
Agreed on all of the above with the addition of LOTS of cutting oil to keep your cutting edge in as good of shape as possible for as long as possible. If the hole is so deep that you need to peck, it's trouble. If you manage to hit that magic feed/speed where it keeps a chip cutting, keep feeding don't stop!
Cutters and drill bits with a bit of cobalt in them can help. I recently had a troublesome SS bolt to drill out that was already work hardened by a friend. TiN coated endmills were building up the stainless on the cutting edge and generally performing miserably. The cobalt drill screamed, but it got through the whole job, edge intact, with a fair amount of feed pressure. I would have liked to find out what a carbide endmill did to it, but the job was done.
One big tip I recently discovered is while TiN coatings (the goldish coating) are great for avoiding aluminum buildup on the cutting edge, avoid TiN coating on troublesome stainless , because the coating causes a less sharp edge because of the thickness of the coating. I've yet to do a side by side comparison, but it sounds good to me and my uncoated cobalt bits have worked well for me. Skip the coated, and use lots of oil, and feed hard, and spin slower.
13mm deep isn't deep enough to need pecking, that's good.
In order of increasing annoyance:
Got a carbide 16mm center cutting endmill (slot drill)? Lots of cutting oil and finish it off with the endmill in the drill chuck.
Or a cobalt 16mm will probably do the trick, but don't be surprised if you need a lot of pressure to get it going.
And lastly, if all you have is HSS drills, I'd try a smaller diameter drill bit, but still large enough to not be too terribly fragile and not so large that it'll cause tip chipping on the 16mm when you go back to it. Perhaps 5-6mm. Make sure it's dead sharp, slow down the RPM, and push hard to start it. Sharpen and repeat with the 16mm.