Either pony up the cash for a
real power auger:
Or pony up the cash for a drain cleaning service.
Warning? A real auger (in inexperienced hands) can potentially damage "buried things" that will be very expensive to fix. If your buired drainage is cast iron (which I'm assuming it is, given the date of your house),
rust scale may well be a large part of the problem; these kinds of blockages require use of a proper auger. The little handyman "snake" is only useful for hair balls and "soft" clogs.
Was your sink (at one time) draining properly? or has it always been "slow"?
There are a few other potential problems (venting, for one) that can also
contribute to such a situation. The vent termination on a lot of kitchen sinks is quite low down (near the eave overhang of the roof) and after many years, it can get blocked by an accumulation of fallen leaves and "tree stuff". This is a potential "contributor" to a slow draining sink. I've seen it many times, usually where there are a couple of 90's used (around a window opening) to get the vent pipe into the sink cabinet. The horizontal branch on the vent is sometimes
packed solid with "tree stuff". There are also other things (right under the kitchen sink) that may be "contributors", but it's hard to say from here...
I've busted out lots of concrete and cast drainage over the years (adding/moving bathrooms in basement developments) and auger everything I can get access to before closing back up. The kitchen line is almost always the worst, especially if the drop into the floor is located a long way from the main stack(s)/building drain, (which is often the case). There
should be (required by code) an accessible cleanout located at the point where the "stack" from your kitchen sink enters the concrete floor. This is usually done with a 45 "TY" fitting. The soil pipe enters the branch, and a cleanout plug is located in the upstream end of the fitting. A 45 fitting on the other end (under the floor) is used to "roll out" to the proper gradient (1":4') for the trip to the stack or building drain.
Personally? Given the age of your house, It's time to call a plumber...
...otherwise, you'll most likely be dealing with this as an ongoing issue for years to come.
I'd pop by for a look...but I live in St. Albert.