Great question, difficult to answer.
There's a great deal of variation in the CO2 footprint of manufactured goods. So we can compare many different goods, I suggest we use Kg CO2 per £1000 cost of product. To get an idea of the range of footprints here are a few datapoints:-
720 Kg CO2 per £1000 So a £4000 car has a manufacturing CO2 footprint of roughly 2.88 tonnes
79 Kg CO2 per £1000 This include transport costs
A4 paper 564 Kg CO2 per £1000
From this you can scale it down so that a cheap £5 toaster has a CO2 footprint of between 0.4Kg and 3.6Kg of CO2.
This energy use calculator suggests that an average toaster would use 0.24 kWh per Day, the CO2 intensity of UK electricity generation is 0.316 KgCo2/kWh, which means that a broken toaster like yours might use around 0.076 Kg of carbon per day.
It would take between 5 and 48 days to recoup the CO2 from manufacturing.
You might want to spend more money on a fancy toaster rather than a £5 one, that's up to you, in terms of CO2 it will take between a few days and two months of daily use to pay off the CO2 debt.
Whether its worth repairing the toaster depends on how much you value your time, how much you value your toaster and the price you put on CO2.
I usually use a CO2 price of £10 per tonne, which is £0.01 per Kg, almost insignificant in terms of the CO2 footprint of manufacturing and use of one toaster. My point here is that the CO2 price of a manufactured good is a tiny fraction of its retail price tag. The CO2 footprint of an expensive toaster is effectively the same as the CO2 footprint of a cheap toaster. Even if we used a £100 per tonne CO2 price its still next to nothing compared to the retail price.
Suppose your labour time is worth £15 per hour, if it takes you more than 20 minutes to repair your £5 toaster then its more efficient to throw it away and buy a new one on Amazon, and spend a few second's worth of labour time to pay for the CO2. Although Amazon does list toasters for up to £2,000, that might be a better case for repairing rather than discarding (is toaster insurance a thing?)