Is there anything, that doesn't seem completely suicidal, that a landsknecht could have done, were he to meet a warrior, in full plate armour, on the battlefield?
Generally, armour works, which is why people were willing to go to the expense and weight of wearing it. However, this doesn't mean that armoured soldiers were invulnerable.
One very common threat on the battlefield was the common missile weapons: arrows, crossbow bolts, and bullets. Arrow, bolts, or bullets that penetrated the chest or abdominal cavity or the skill were likely to cause death - if not immediately, then later through infection. Arrow and crossbow bolt wounds to the arms and legs were much less likely to kill, and were also less likely to be incapacitating. Ideally, one's armour would provide all-over protection against arrows and bolts, but since archers and crossbowmen went to the trouble of using high draw weight weapons, potentially delivering arrows/bolts with over 100J of energy to the target, the thickness of iron required to reliably stop them is too great. That is, if you were to make iron armour that was all-over arrowproof, it would be too heavy. The most important parts of the body to protect from arrows and bolts are the parts when they are most likely to cause fatal wounds: the head and torso. Thus, we find much plate armour where the breastplate and the helmet (or at least the frontal parts of the helmet) are 2-3mm thick. This is enough to be reasonably reliable for stopped arrows and bolts, and in the early 16th century was also good protection against guns.
A complete plate armour that was 2-3mm thick iron/steel, for all parts of the armour, would be too heavy. It would particularly burden the arms and legs, since they need to move more than the head and torso, and the armour on the arms and legs needs to be moved more. The usual recipe for armour is then torso and head armour of about 2-3mm (at least for the parts most likely to be hit), while arm and leg armour is much thinner. Arm and leg armour was often under 1mm thick; 0.8mm to 1mm was common. An arrow or bolt striking square would have a good chance of penetrating it, but if it hits at an angle, it might glance off. This thinner armour, 0.8-1mm thick, still provides good protection against many weapons. It is good protection against swords - it might be dented, but it will not be cut through.
Even without gaps (e.g., the backs of the thighs are often unprotected in equestrian armours, and the common gaps in joints, palms of hands, and groin), the Landsknecht of your question has a chance. The best tactics would depend on what the knight is armed with. If the knight has a shorter weapon, your Landsknecht could try to stay out of reach, and use his long messer to target, for example, the knight's hands - protected by gauntlets, but the gauntlets are usually the thinnest part of the armour, to minimise weight on the hands. A wiser tactic would be to find friends - on the typical battlefield of the time, the Landsknecht and his comrades would outnumber the knights. Rather than fight an armoured knight one-on-one, it would be safer to make it 3-vs-1 or 5-vs-1. Let us consider the effectiveness of two common Landsknecht weapons: the arquebus and the halberd.
Arquebus: Just as arrows and bolts can penetrate the thinner armour on arms and legs, so can an arquebus ball. At close range, an arquebus would have a good chance of penetrating the thinner breastplates and helmets (i.e., if they are only about 2mm thick). An arquebus of the time would fire with a muzzle energy of approximately 1500J, which would have a good chance of penetration 2mm of iron. At longer ranges, the energy of the ball falls off, and a breastplate would probably stop it. However, even at longer ranges, arm and leg armour could be penetrated. At close range, even striking at an angle, a ball would probably penetrate arm and leg armour. Compared to an arrow/bolt, the arquebus is much more likely to cause an incapacitating wound to an arm or leg - while an arrow or bolt will make a not-too-huge cut or hole, and be stopped if it hits the bone, an arquebus ball will rip a larger hole clean through, and break the bone if it hits.
Halberd: While 0.8-1mm armour is difficult to cut, and offers good protection against swords (no cut, although denting of the armour is to be expect), polearms can deliver more energy. The mass of the polearm (about 2.5-3kg for a halberd of the time) means that a thrust with the point has a lot of energy, and is likely to penetrate thin armour. Swung, the back spike should easily punch through such armour, and even the axe blade could cut through (and if it doesn't, it will dent the armour so badly that a sword would be quite jealous). For a demonstration of the potential effectiveness of a halberd against arm and leg armour, consider this test:
where the back spike of a halberd is driven through 1.5mm mild steel, much thicker than typical arm and leg armour, and perhaps about twice as protective. If it can do this to something twice as protective as arm armour, what will it do to the arm of an armoured knight?
Of course, this doesn't mean that your Landsknecht would have an easy time of it - the knight is still better-protected than the Landsknecht, and isn't likely to just stand there cooperatively to be shot or struck by halberds. But a Landsknecht would have a chance, and given enough comrades, would even have a good chance.
References:
For penetration of armour, muzzle energy, and thickness of armour, see
For the relative lethality of arrow wounds to head and torso and limbs, see my answer in https://www.reddit.com/r/AskHistorians/comments/ba42vw/death_by_arrow_really/