The decays B+ → J/ψπ+π−K+ are studied using a data set corresponding to an integrated luminosity of 9 fb−1 collected with the LHCb detector in proton-proton collisions between 2011 and 2018. Precise measurements of the ratios of branching fractions with the intermediate ψ2(3823), χc1(3872) and ψ(2S) states are reported. The values are BB+→ψ2(3823)K+×Bψ2(3823)→J/ψπ+π−BB+→χc1(3872)K+×Bχc1(3872)→J/ψπ+π−=(3.56±0.67±0.11)×10−2,BB+→ψ2(3823)K+×Bψ2(3823)→J/ψπ+π−BB+→ψ(2S)K+×Bψ(2S)→J/ψπ+π−=(1.31±0.25±0.04)×10−3,BB+→χc1(3872)K+×Bχc1(3872)→J/ψπ+π−BB+→ψ(2S)K+×Bψ(2S)→J/ψπ+π−=(3.69±0.07±0.06)×10−2, where the first uncertainty is statistical and the second is systematic. The decay of B+ → ψ2(3823)K+ with ψ2(3823) → J/ψπ+π− is observed for the first time with a significance of 5.1 standard deviations. The mass differences between the ψ2(3823), χc1(3872) and ψ(2S) states are measured to be mχc1(3872)−mψ2(3823)=47.50±0.53±0.13MeV/c2,mψ2(3823)−mψ2(2S)=137.98±0.53±0.14MeV/c2,mχc1(3872)−mψ2(2S)=185.49±0.06±0.03MeV/c2, resulting in the most precise determination of the χc1(3872) mass. The width of the ψ2(3823) state is found to be below 5.2 MeV at 90% confidence level. The Breit-Wigner width of the χc1(3872) state is measured to be ΓBWχc1(3872)=0.96+0.19−0.18±0.21MeV which is inconsistent with zero by 5.5 standard deviations.