Knives don't stick into ice well.
You are also looking at the problem of getting them out and the risk of cutting your hand open when you attempt to stab them through the ice.
Better is a couple of good sized pieces of dowel.
About as big around as a broom or shovel handle and long enough to fit your hand well, about five inches long is good.
You drill a hole into the end length ways and put a hard nail into them.
It only needs to poke out about an inch or so, maybe an inch and a half, if it is too long it will just bend or break.
The hole you pre-drill for the nail is just to prevent splitting the dowel when you drive the nail into it.
If the hole is bigger than the nail you can epoxy the nail into it too.
Most people cut the nail head off and sharpen them a bit, they don't have to be super needle sharp though.
To prevent losing them drill another crosswise hole in the other end. Then run a string through your sleeves and across your shoulder just like you were going to tie mitts on and tie the ice picks to it.
You can likely tuck them just inside the cuffs of your sleeves while paddling, but make sure they stay easy to get out of your sleeves. (maybe leave enough string hanging out to grab.)
You might be able to tie them to a vest.
They are not very bulky.
Really just two round sticks about 4 or 5 inches long with short nails sticking out one end of them.
The main thing is to keep them easy to reach.
The idea is to hold them in your fist and slam them into the ice like a hammer blow.
They should give you enough of a grip to pull yourself out onto the ice and slide yourself towards shore like a seal does, until the ice is thick enough to crawl on and then to walk on.
(Don't hurry to stand up on the ice. Wiggling and crawling is better than falling through thin ice. It would be very bad to fall through the ice after getting out once)
If the ice is really thin at the edge you might end up breaking it off until you get to where it is thick enough to hold the nail when you punch it through the ice.
(to bad this is unlikely to practiced before you really need to do it. If you can get close to thin ice safely practice punching the nails through it. You will find out how thick it has to be to hold a nail)
I just found a video that might help explain it a bit more.
self rescue