I have several vertical radiators and they work well and do heat all the way way up. Heat does rise, by the way! . You must calculatre the heat output you need for the space you want to heat and put in a sufficiently sized radiator (or an even higher heat output). A tiddly little one will not be sufficient and the finish of the radiator can reduce heat output. For example chrome ones have less output than standard painted ones. There is no difference in heat output whether it is vertical or horizontal.
I would suggest that you also look at a supplementary heating system in the dining area if it is large, but if the kitchen/dinaing area is the same size as your lounge, and you only hasve one radiator in there, then there should not be a problem, assuming you have an insulated house of course.
Yes, be sure you check the BTUs you need (there are various online calculators that factor in size of room, how much glass there is, whether there is another room above/ below it, how many outside walls it has etc) and get the white ones rather than chrome (engineer friend told us that they retain heat better once the heating goes off). We have 2 in a very large kitchen diner and they're very effective. I'd recommend the brand Acova, they get very good reviews.
I have 6 of them in an old victorian and they work very well. I got column style and made sure i added to the required btu,s needed and they have thermostatic valves on each one.
House is very warm now and you can turn them down very easily. Will put them in again when i move and wish i could be bothered to rip them out and take them with me but not worth the time or hassle however much i love them. Not sure about john and effectiveness, ive found them more effective as take up less useful space in a room than a horizontal radiator and heat just as well.