I made the mistake of not rehydrating. For aeration I shake up the bucket for a minute or so and then pitch.
Normally the cell count in a dry yeast packet is plenty, but when you sprinkle it directly into wort you kill about half of the cells so you are effectively underpitching. The cell walls burst because they are not meant to be filled with wort. Rehydrate with warm water (about 105) and a pinch of yeast nutrient. There's really no good reason to skip this simple step.
Aeration is your second problem. To quote "Yeast", by Chris White: 'Using proper levels of dissolved oxygen is just as important as using proper pitching rates (number of yeast cells). Lack of oxygen cause many fermentation problems. Stuck fermentations, long fermentation times, under-attenuated beers, yeast stress, and off flavors are often the result of too little oxygen". The proper amount is 8 to 10 PPM (parts per million). Since there's only 8 ppm in the air it's impossible to get any higher than 8 ppm using air. In tests they ran 5 minutes of shaking and splashing only resulted in 2.7 ppm, and you did even less than that. So basically you killed half the yeast and didn't give the survivors a healthy environment to do their work.
You only really have 2 options:
1- Give it a few weeks and hope for the best.
2- Make a starter and pitch more actively fermenting yeast
without adding additional oxygen. Oxygen at this stage will lead to bad tasting, oxydized beer. This doesn't happen normally because the yeast absorb all the oxygen
before (lag phase) they begin to ferment.
I'd recommend option 1 for you because a your first yeast starter shouldn't be done under the stress of "saving" a beer. Also, if you give it a few weeks to age on what yeast are in there they clean up much of the off flavors and it might turn out alright. Next time, if you rehydrate your yeast and aerate
much more, you beer will definitely turn out better...