Dried orange peel lights easily and burns really well - may just do the job you need, lighting from the newspaper and lighting the kindling. When I eat an orange, the peel goes on top of the stove to dry.
Silage net also burns extremely well. If you don't have your own, your local farmer will almost certainly let you have some.
Our house used to be cold and damp too, before we got the Rayburn. But by keeping an old newspaper, and some twigs and small sticks near to the woodburner, there is always some dry fire-starting stuff.
My fire-starting technique is:
* empty the grate - clear the route you want the air drawn through to the underneath of the fire. I agree with whoever said that with some woodburners it actually helps to not clean out the part-burned stuff and ash from on top of the grate completely, just where you are building the start of the fire, and underneath where the air is to be drawn in
* make a draft proof 'pit' if necessary (that remaining ash and so on can help here, or a few logs as FW says)
* scrumple three sheets of newspaper loosely and place in bottom of pit
* twist two separate sheets of newspaper into tightly-twisted sticks and place on top
* if I have some, shred a couple of old loo rolls or other useful cardboard / stiff paper type stuff on top - just a bit, don't smother it
* a dried orange peel next, if I have one
* three or four small twigs, must be very dry, dropped on top, spread out but crossing over each other
* three or four slightly larger very dry twigs / sticks on top of those, again spread out but crossing each other
* a couple of bits of chopped pallet or similar on top of these - if sticks, about 3/4" diameter - and
dry* light the newspaper at the bottom in a couple of places
* manage the draft to ensure the air is being drawn up through the fire from underneath. In an open fireplace, this is usually accomplished by a sheet of a broadsheet paper opened up and held across the whole fireplace, so that the only air gap is below the grate. Hold it taut so it doesn't get sucked in and lit! Or a big sheet of cardboard that covers the whole opening works brilliantly - and once it's started to be a bit scorched, it's great to be torn up for layer #3 next time
* when the twigs start to crackle, put a bit more wood on top - don't smother it
If I have some silage net and want to use it, it goes on top of the first layer of twigs with some more twigs on top of it / stuck through it. (Of course the net needs to be dry too.)
Other tips I've picked up over the years... save the 'brash' from larch and other coniferous trimmings, keep them dry and in two years they are phenomenal firelighting material - when they're so dry they crumble.
Some woodworking places will sell you - or let you collect - sawdust and wood planings/chippings; as long as these are dry, a handful in a paper bag is good firelighting/rescuing stuff.
Our local woodworking place produces discs of dried, compressed wood waste; £3 a bag, and these are as good as any firelighter once lit. (Not
quite as easy as Zip to light, but not very hard.)