Don't make the Burning Blade until you can reach 16 fire from the start of battle.
Smithing For Dummies on R4 helps to make the sword.
If you get a Thorn Whip before R9, consider not selling the Broom until R10 in case you see Hardwood. If you don´t see Hardwood, switch to Flame Whip.
Also, Flame can be swaped with chipped gems Same thing with Burning Coal and flawed+ gems.