How to connect 5 kW electric heater to the weak wiring in house?
Running double the current through your wiring causes 4 times the dissipation in it, and so more or less 4 times the temperature rise (assuming conduction and natural convection as the principal sources of cooling at these temperatures). It's the temperature rise that causes the damage, softening insulation, starting fires etc.
Using the triac circuit is supplying power for 50% of the time. Although that would cut your heater dissipation in half, it only cuts your '4 times too much' wiring dissipation in half as well, leaving the wiring temperature still double what it should be. Given that the existing '10 A' wiring is rated for a significant temperature rise, even twice the rise is still far too hot for safe operation of the plastic insulation.
Insulation relies on being solid. At double the rated temperature rise above ambient, it's likely to soften, and flow away from anywhere there's pressure on it. This could cause conductors to contact each other. In the best case, the short would open the circuit breaker. You'd then have a dead circuit that would have to be repaired/rewired by an electrician. In the worst case, an arc between the conductors could start a fire.
Your only two sensible options are to rewire, or to buy a much lower power heater. Or three options, stop using the heater. Or four options, continue using the heater, and have a good story ready for the fire investigators.
Or the fifth option, install a 2:1 step down transformer at the sending end of your 10 A feeder. This will run the wiring at 10 A, so the wiring will run at rated, but the heater will run at 1.25 kW. It might be that buying a suitably sized transformer is competitive with hiring an electrician to run a 20 A cable. It will also only work if this is an isolated feeder to only your heater, due to the non standard voltage.
What about the 6th option of using the triac to turn on for less than 1/2 a cycle? That's possible. If you turn the triac on for the last 1/3rd of the cycle, then you'll deliver about half the rms current. The heater and wiring will dissipate 25% of their 'full' power. You don't need to worry about the peak current, just the rms aka heating effect, at least for modest peaks like this.