For fusion to occur we need extremely high temperature at the core to overcome coulomb barrier or for the nucleus to be close enough to tunnel into each other. Is it possible for type 2 civilization to develop advanced tech that can balance the inward pressure (gravity + magnetic field) and outward pressure (fusion of course!) to turn Jupiter into giant thermonuclear furnace? If Jupiter becomes a micro-star how much light does Earth receives from it minus the Sun? Is it safe to bath under Jovian light?
Take a look at this question. Apparently Jupiter doesn't have enough mass to sustain a fusion reaction; it requires a lot more mass even for a small star. If heated enough it would even lose mass as atoms reached escape velocity.
Assuming your type-2 civilization started a fusion reaction, the fusion wouldn't continue unless they actively kept it going, since fusion occurs in stars due to exposure of the fusing atoms to heat (from energy of prior fusion) and pressure (due to the star's gravity, requiring enough mass).
A star with the mass of Jupiter (I won't say size, since it would expand if heated) would also not be very bright and would radiate in the red to infrared (heat).
I suppose your type 2 civ might put a handwavium device in Jupiter's orbit that sucked hydrogen from it and made a bright mini-sun from fusing that hydrogen. Since the device is handwavium in design, you could specify whatever radiation levels and types you desire.
You ask very specifically «…can balance the inward pressure … and outward pressure … to turn Jupiter into giant thermonuclear furnace?»
I think no, as Mark’s answer explains.
But if you suppose that you want the result of a really tiny star but don’t care whether it’s done by adding pressure artificially and letting nature take its course, or in some completely different manner, you have a lot more flexibility.
In general, you don’t want to perform fusion at the core. Check back in in a year to find out whether ♃ even has hydrogen at its core!
You could have a huge number of fusion devices floating in the atmosphere, which en masse would make the surface appear to glow.
The light from these devices can be directed specifically toward the moons you wanted to light up. So standing on that moon you would see a bright sun of the desired total brightness, but from any other vantage point you would see a planet not glowing!
No. Just think about brown dwarfs. These objects are substellar objects with a mass of 15 to 75 Jupiter masses. But they still have not enough mass to start the fusion. This means, if you want to transform Jupiter into an actual star you need to increase Jupiters mass by more than 75 times its actual mass.
Since the specifications are for a KII civilization to make Jupiter into a star, then it is possible using some ultra science.
What they will have to do is use "Catalytic Fusion" to allow Jupiter to begin fusion reactions without the mass and pressure needed to create a real star (I believe that even a "Brown Dwarf" needs to be between 4 to 12X or more more massive than Jupiter to get an anemic fusion reaction going). Sending an intense beam of Muons towards the Jovian core is the key to Catalytic Fusion. The Muons hit the metallic hydrogen layer and displace the electrons, but since they are 200X more massive than an electron they form atoms which are 200X smaller, bringing the nuclei close enough to overcome the coulomb barrier and initiate fusion.
The reason we don't have these sorts of reactors on Earth is simply that creating a muon beam is energy intensive and the Muons decay in a very short time span, initiating far too few fusion reactions to generate a net positive energy output.
For a KII civilization, this issue may well have been overcome, and low energy means of generating muon beams might be quite common. If that is the case, then sustaining the beam aimed at Jupiter isn't a problem and the resulting fusion reaction is a net positive, releasing the light and heat energy that they desire (presumably to terraform the moons of Jupiter into oceanic habitats).
The effects of the Jupiter Star on the rest of the Solar System will depend on what exactly the KII civilization is like; one with no space infrastructure between Jupiter and Earth will see something much like the epilogue of 2010: Odyssey Two, with illumination similar to moonlight when the New Sun is in the night sky. If the KII civilization has enclosed the sun in a Dyson swarm @ 1 AU in diameter, then people within will not see the New Sun at all. It is also possible that the KII civilization will enclose the New Sun in a Dyson swarm of its own, to trap the energy and divert some to power the Muon beam which makes it possible.